Chapter 7 Credit Rating Systems

Page 259 Chapter 7 Credit Rating Systems 1 Introduction In this chapter we explore the traditional and prevalent approach to credit risk assessment the credit rating system. Most rating systems are based on both quantitative and qualitative evaluations. The final decision is based on many different attributes, but usually it is not calculated using a formal model that would show how to weight all these attributes in a normative way. In essence, the systems are based on general considerations and on experience, and not on mathematical modeling. They cannot therefore be regarded as precise, and they also clearly rely on the judgement of the ratings evaluators. Ratings systems are usually applied to nonfinancial corporations, as special approaches are employed for banks and other financial institutions. First of all, we describe the rating systems of the two main credit rating agencies, Standard & Poor's and Moody's. Almost all of the public issues of debt instruments in the United States and Canada are rated by these agencies. Their ratings of public issues are made available to the public, as are the periodic revisions of these ratings. Companies and instruments are classified into discrete rating categories that correspond to the estimated likelihood of the company failing to pay its obligations. In Section 3 we show how an internal rating system in a bank can be organized in order to rate creditors systematically. It should

Page 260 again be emphasized that while this system is based on the extensive experience of a commercial bank, it is not based on any normative model. Other banks may have somewhat different systems, but most are of a similar nature. In Sections 5 to 7 the details of the rating process and considerations are described. The main problem faced by banks is obtaining information about companies that have not issued traded debt instruments. The data about these companies are of unproven quality and are therefore less reliable, and it can be a challenge to extract the minimum required information in order to improve the allocation of credit. The credit analysts in a bank or a rating agency must take into consideration many attributes of a firm: financial as well as managerial, quantitative as well as qualitative. The analysts must ascertain the financial health of the firm, and determine if earnings and cash flows are sufficient to cover the debt obligations. The analysts would also want to analyze the quality of the assets of the firm and the liquidity position of the firm. In addition, the analysts must take into account the features of the industry to which the potential client belongs, and the status of the client within its industry. The effects of macroeconomic events on the firm and its industry should also be considered, as well as the country risk of the borrower. Combined industry and country factors can be assessed to calculate the correlation between assets for the purpose of calculating portfolio effects. (Chapter 8 discusses portfolio effects with regard to credit risk.) In a very schematic way, Figure 7.1 illustrates the environment of the borrower that the credit analyst must assess in order to determine the creditworthiness of the borrower and thus the interest spread that the bank should charge. A major consideration in providing a loan is the existence of a collateral, or otherwise of a loan guarantor, and the quality of the guarantee. This issue of guarantee is especially important for banks providing loans to small and medium-sized companies that cannot offer sufficient collateral. In order to be reliable, the classification method must be consistent over time and be based on sound economic principles. The approach presented here to evaluate credit risk is consistent with the new directives of the BIS to implement a systematic procedure for credit risk assessment. This is important because the output of the rating procedures is a key determinant when evaluating the probability of default

Page 261 Figure 7.1 The Environment of the Borrower and the loss given default, which in turn are key inputs in credit VaR calculations. As we have discussed in earlier chapters, credit VaR calculations are used to determine the amount of capital that the bank should allocate against its exposure to credit risk. For example, the loss distribution and capital assessment may be based on the RiskMetric's CreditMetrics model (originally developed by JP Morgan), which uses as inputs the probability that each obligor migrates from a certain rating to another rating over a one-year period, and the loss given default for each facility (see Chapter 8). 2 Rating Agencies 2.1 The External Agency Rating Process The issuance of bonds by corporations is a twentieth-century phenomenon. It started at the beginning of the century, at approximately the same time that the first papers and articles were published on the analysis of accounting ratios, as a means of diagnosing the financial strength of a company. By the 1920s, this approach had been commercialized and specialized firms were offering their services, and promoting the merits of ratio analysis. This was also the period when Moody's (1909),

Page 262 Standard & Poor's (1916), and other agencies started to rate public debt issues. Over the last 30 years, the introduction of new financial products has led to the development of new methodologies and criteria for credit rating: Standard & Poor's (S&P) was the first rating company to rate mortgagebacked bonds (1975), mutual funds (1983), and asset-backed securities (1985). A credit rating is not, in general, an investment recommendation concerning a given security. In the words of S&P, "A credit rating is S&P's opinion of the general creditworthiness of an obligor, or the creditworthiness of an obligor with respect to a particular debt security or other financial obligation, based on relevant risk factors." 1 In Moody's words, a rating is, "an opinion on the future ability and legal obligation of an issuer to make timely payments of principal and interest on a specific fixedincome security." 2 "Moody's ratings of industrial and financial companies have primarily reflected default probability, while expected severity of loss in the event of default has played an important secondary role. In the speculative-grade portion of the market, which has been developing into a distinct sector, Moody's ratings place more emphasis on expected loss than on relative default risk." 3 Since S&P and Moody's are considered to have expertise in credit rating and are regarded as unbiased evaluators, their ratings are widely accepted by market participants and regulatory agencies. Financial institutions, when required to hold investment-grade bonds by their regulators, use the ratings of credit agencies such as S&P and Moody's to determine which bonds are of investment grade. The subject of a credit rating might be a company issuing debt obligations. In the case of such "issuer credit ratings," the rating is an opinion on the obligor's overall capacity to meet its financial obligations. The opinion is not specific to any particular liability of the company, nor does it consider the merits of having guarantors for some of the obligations. In the issuer credit rating category are counterparty ratings, corporate credit ratings, and sovereign credit ratings. Another class of rating is "issue-specific credit ratings." In this case, the rating agency makes a distinction, in its rating system and symbols, between long-term and short-term credits. The shortterm ratings apply to commercial paper (CP), certificates of deposits

Page 263 (CDs), and put bonds. 4 In rating a specific issue the attributes of the issuer, as well as the specific terms of the issue, the quality of the collateral, and the creditworthiness of the guarantors, are taken into account. The rating process includes quantitative, qualitative, and legal analyses. The quantitative analysis is mainly financial analysis and is based on the firm's financial reports. The qualitative analysis is concerned with the quality of management, and includes a thorough review of the firm's competitiveness within its industry as well as the expected growth of the industry and its vulnerability to technological changes, regulatory changes, and labor relations. Figure 7.2 illustrates the process of rating an industrial company. The process works through sovereign and macroeconomic issues, industry outlook and regulatory trends, to specific attributes (including quality of management, operating and financial positions), and eventually to the issuespecific structure of the financial instrument. When rating a company, the nature of competition within its industry is a very important consideration. In trying to illustrate Figure 7.2 Moody's Rating Analysis of an Industrial Company

Page 264 its evaluation process, S&P uses an example of a firm from the airline industry. For such a firm, the analysis concentrates on issues such as market position in specific markets locally and internationally, including barriers to entry, revenue generation (including pricing, utilization of capacity, service reputation, and productivity), cost control (for labor, fuel, commissions), and the quality of the aircraft fleet. The assessment of management, although subjective in nature, investigates how likely it is that it will achieve operational success, and its risk tolerance. The rating process includes meetings with the management of the issuer to review operating and financial plans, policies, and strategies. All the information is reviewed and discussed by a rating committee with appropriate expertise in the relevant industry, which then votes on the recommendation. The issuer can appeal against the rating before it is made public by supplying new information. The rating decision is usually issued four to six weeks after the agency is asked to rate a debt issue. Usually the ratings are reviewed once a year, based on new financial reports, new business information, and review meetings with management. A "credit watch" or "rating review" notice is issued if there is reason to believe that the review may lead to a credit rating change. A change of rating has to be approved by the rating committee. The rating process of S&P is described in Figure 7.3. (An almost identical process is used by all rating agencies.) 2.2 Credit Ratings by S&P and Moody's Standard & Poor's (S&P) is one of the major rating agencies in the world, operating in more than 50 countries. Moody's operates Figure 7.3 Standard & Poor's Debt Rating Process

Page 265 mainly in the United States but has many branches internationally. Moody's and S&P have a dominant position to such an extent that U.S. Justice Department inquiries have considered whether there may be "anticompetitive practices" in the bond rating industry. 5 Table 7.1 and Table 7.2 provide the definitions of the ratings categories of S&P and Moody's for long-term credit. We also show in Table 7.3a and Table 7.3b the short-term ratings of S&P and Moody's, respectively. Moody's short-term debt ratings employ three designations only, all judged to be investment grade. If we focus on S&P (Table 7.1), we can see that the symbols are identical for issue and issuer credit ratings, and also that the definitions closely correspond to one another. The categories are defined in terms of default risk and the likelihood of payment for the issuer. Issues rated in the four highest categories (i.e., AAA, AA, A, and BBB of S&P and Aaa, Aa, A, and Baa of Moody's) are generally considered as being of investment grade. Some financial institutions, for special or approved investment programs, are required to invest only in bonds or debt instruments that are of investment grade. Obligations rated BB, B, CCC, CC, and C (Ba, B, Caa, Ca, and C of Moody's), are regarded as having significant speculative characteristics. BB (Ba of Moody's) is the least risky and C is the most risky within the speculative grade category. As can be seen in Tables 7.1 and 7.2, the rating categories used by S&P and Moody's are quite similar, though differences of opinion can lead in some case to different ratings of specific debt obligations. Moody's applies numerical modifiers 1, 2, and 3 in each generic rating classification from Aa through Caa. The modifier 1 indicates that the obligation ranks in the higher end of its generic rating category; the modifier 2 indicates a mid-range ranking; and the modifier 3 indicates a ranking at the lower end of that generic rating category. For example, B1 in Moody's rating system has an equivalent ranking to B+ in S&P's rating system. 2.3 The Differences in Ratings While the rating agencies use similar methods and approaches to rate debt, they sometimes come up with different ratings of the same debt investment. In their studies of the credit rating

Page 266 TABLE 7.1 S&P Ratings Category Definitions AAA AA A BBB BB B CCC CC C D An obligation rated AAA has the highest rating assigned by Standard & Poor's. The obligor's capacity to meet its financial commitment on the obligation is extremely strong. An obligation rated AA differs from the highest rated obligations only in small degree. The obligor's capacity to meet its financial commitment on the obligation is very strong. An obligation rated A is somewhat more susceptible to the adverse effects of changes in circumstances and economic conditions than obligations in higher rated categories. However, the obligor's capacity to meet its financial commitment on the obligation is still strong. An obligation rated BBB exhibits adequate protection parameters. However, adverse economic conditions or changing circumstances are more likely to lead to a weakened capacity of the obligor to meet its financial commitment on the obligation. An obligation rated BB is less vulnerable to nonpayment than other speculative issues. However, it faces major ongoing uncertainties or exposure to adverse business, financial, or economic conditions which could lead to the obligor's inadequate capacity to meet its financial commitment on the obligation. An obligation rated B is more vulnerable to nonpayment than obligations rated BB but the obligor currently has the capacity to meet its financial commitment on the obligation. Adverse business, financial, or economic conditions will likely impair the obligor's capacity or willingness to meet its financial commitment on the obligation. An obligation rated CCC is currently vulnerable to nonpayment, and is dependent upon favorable business, financial, and economic conditions for the obligor to meet its financial commitment on the obligation. In the event of adverse business, financial, or economic conditions, the obligor is not likely to have the capacity to meet its financial commitment on the obligation. An obligation rated CC is currently highly vulnerable to nonpayment. The C rating may be used to cover a situation where a bankruptcy petition has been filed or similar action has been taken, but payments on this obligation are being continued. The D rating, unlike other ratings, is not prospective; rather, it is used only where a default has actually occurred and not where a default is only expected. Standard & Poor's changes ratings to D either: On the day an interest and/or principal payment is due and is not paid. An exception is made if there is a grace period and S&P believes that a payment will be made, in which case the rating can be maintained; or Upon voluntary bankruptcy filing or similar action. An exception is made if S&P expects that debt service payments will continue to be made on a specific issue. In the absence of a payment default or bankruptcy filing, a technical default (i.e., covenant violation) is not sufficient for assigning a D rating. + or The ratings from AA to CCC may be modified by the addition of a plus or minus sign to show relative standing within the major rating categories. R The symbol is attached to the ratings of instruments with significant noncredit risks. It highlights risks to principal or volatility of expected returns which are not addressed in the credit rating. Examples include: obligations linked or indexed to equities, currencies, or commodities; obligations exposed to severe prepayment risk such as interestonly or principal-only mortgage securities; and obligations with unusually risky interest terms, such as inverse floaters. Source: Reproduced from Corporate Ratings Criteria of S&P for 1998.

Page 267 TABLE 7.2 Moody's Rating Category Definition Aaa Aa A Baa Ba B Caa Ca C Bonds which are rated Aaa are judged to be of the best quality. They carry the smallest degree of investment risk and are generally referred to as ''gilt edged." Interest payments are protected by a large or by an exceptionally stable margin and principal is secure. While the various protective elements are likely to change, such changes as can be visualized are most unlikely to impair the fundamentally strong position of such issues. Bonds which are rated Aa are judged to be of high quality by all standards. Together with the Aaa group they comprise what are generally known as high-grade bonds. They are rated lower than the best bonds because margins of protection may not be as large as in Aaa securities or fluctuation of protective elements may be of greater amplitude or there may be other elements present which make the long-term risk appear somewhat larger than the Aaa securities. Bonds which are rated A possess many favorable investment attributes and are to be considered as upper mediumgrade obligations. Factors giving security to principal and interest are considered adequate, but elements may be present which suggest a susceptibility to impairment some time in the future. Bonds which are rated Baa are considered as medium-grade obligations (i.e., they are neither highly protected nor poorly secured). Interest payments and principal security appear adequate for the present but certain protective elements may be lacking or may be characteristically unreliable over any great length of time. Such bonds lack outstanding investment characteristics and in fact have speculative characteristics as well. Bonds which are rated Ba are judged to have speculative elements; their future cannot be considered as wellassured. Often the protection of interest and principal payments may be very moderate, and thereby not well safeguarded during both good and bad times over the future. Uncertainty of position characterizes bonds in this class. Bonds which are rated B generally lack characteristics of the desirable investment. Assurance of interest and principal payments or of maintenance of other terms of the contract over any long period of time may be small. Bonds which are rated Caa are of poor standing. Such issues may be in default or there may be present elements of danger with respect to principal or interest. Bonds which are rated Ca represent obligations which are speculative in a high degree. Such issues are often in default or have other marked shortcomings. Bonds which are rated C are the lowest rated class of bonds, and issues so rated can be regarded as having extremely poor prospects of ever attaining any real investment standing. Source: Moody's Credit Ratings and Research, 1995.

Page 268 TABLE 7.3a The Short-Term Credit Ratings of S&P A-1 A short-term obligation rated A-1 is rated in the highest category by S&P. The obligor's capacity to meet its financial commitment on the obligation is strong. Within this category, certain obligations are designated with a plus sign (+). This indicates that the obligor's capacity to meet its financial commitment on these obligations is extremely strong. A-2 A short-term obligation rated A-2 is somewhat more susceptible to the adverse effects of changes in circumstances and economic conditions than obligations in higher rating categories. However, the obligor's capacity to meet its financial commitment on the obligation is satisfactory. A-3 A short-term obligation rated A-3 exhibits adequate protection parameters. However, adverse economic conditions or changing circumstances are more likely to lead to a weakened capacity of the obligor to meet its financial commitment on the obligation. B C D A short-term obligation rated B is regarded as having significant speculative characteristics. The obligor currently has the capacity to meet its financial commitment on the obligation; however, it faces major ongoing uncertainties which could lead to the obligor's inadequate capacity to meet its financial commitment on the obligation. A short-term obligation rated C is currently vulnerable to nonpayment and is dependent upon favorable business, financial, and economic conditions for the obligor to meet its financial commitment on the obligation. The rating 'D' is given where a short-term debt has actually defaulted. Source: Reproduced from Corporate Ratings Criteria of S&P for 1998. industry Cantor and Packer (1995) show that for 1168 firms rated by both Moody's and S&P at the end of 1993, only 53 percent of the firms rated AA or Aa and AAA or Aaa were rated the same by both agencies. For other investment-grade issues only 36 percent were rated in the same way, while 41 percent of those rated as below investment grade had been awarded the same ratings. Table 7.4 is from Cantor and Packer (1995). It shows the differences between the ratings of the two largest rating agencies, S&P and Moody's, and those of the next two agencies in terms of size and reputation, namely Duff & Phelps and Fitch (which later joined forces with another rating agency, IBCA). The table compares 298 firms rated by Moody's, S&P, and Duff & Phelps and 161 firms rated jointly by Moody's, S&P, and Fitch at year-end 1993. The two smaller agencies, Duff & Phelps as well as Fitch, tend to rate debt issues higher or the same as S&P and Moody's. In only 10 percent or less of the cases did they give a lower rating.

Page 269 TABLE 7.3b Moody's Short-Term Debt Ratings Prime 1 Issuers rated Prime 1 (or supporting institutions) have a superior ability for repayment of senior shortterm debt obligations. Prime 1 repayment ability will often be evidenced by many of the following characteristics: Prime 2 Prime 3 Leading market positions in well-established industries. High rates of return on funds employed. Conservative capitalization structure with moderate reliance on debt and ample asset protection. Broad margins in earnings coverage of fixed financial charges and high internal cash generation. Well established access to a range of financial markets and assured sources of alternate liquidity. Issuers rated Prime 2 (or supporting institutions) have a strong ability for repayment of senior short-term debt obligations. This will normally be evidenced by many of the characteristics cited above but to a lesser degree. Earnings trends and coverage ratios, while sound, may be more subject to variation. Capitalization characteristics, while still appropriate, may be more affected by external conditions. Ample alternate liquidity is maintained. Issuers rated Prime 3 (or supporting institutions) have an acceptable ability for repayment of senior shortterm obligations. The effect of industry characteristics and market compositions may be more pronounced. Variability in earnings and profitability may result in changes in the level of debt protection measurements and may require relatively high financial leverage. Adequate alternate liquidity is maintained. Source: Moody's Credit Ratings and Research, 1995. This issue of ratings differences is an important one. It raises two questions. First, to what extent is the rating quantitatively based and what is the role of judgment? (In Section 3.4 we discuss the measurement of default probabilities and recovery rates.) The second question concerns the independence of the rating agencies. Since the rated companies pay to be rated, there is a perceived danger that business pressures will affect the process. 3 Introduction to Internal Risk Rating In this section we look at an internal risk rating system. A typical risk rating system (RRS) will assign both an obligor rating to each

Page 270 TABLE 7.4 Credit Rating Differences Between Agencies Distribution of Duff & Phelps Ratings Relative to Distribution of Fitch's Ratings Relative to Moody's S&P Moody's S&P Rated higher (%) 47.6 39.9 55.3 46.6 Rated same (%) 42.3 46.5 37.9 43.5 Rated lower (%) 10.1 13.5 6.8 9.9 Average difference in matched rating 0.57 0.16 0.74 0.56 Source: Cantor and Packer (1995), Federal Revenue Bank of New York. borrower (or group of borrowers), and a facility rating to each available facility. A risk rating (RR) is designed to depict the risk of loss 6 in a credit facility. A robust RRS should offer a carefully designed, structured, and documented series of steps for the assessment of each rating. 3.1 Objectivity and Methodology The goal is to generate accurate and consistent risk ratings, yet also to allow professional judgment to significantly influence a rating where this is appropriate. The expected loss is the product of an exposure (say, $100) and the probability of default (say, 2 percent) of an obligor (or borrower) and the loss rate given default (say, 50 percent), in any specific credit facility. In this example, the expected loss is $100.02.50 = $1. A typical risk rating methodology (RRM) initially assigns an obligor rating that identifies the expected probability of default by that borrower (or group) in repaying its obligations in the normal course of business. The RRS then identifies the risk of loss (principal or interest) by assigning an RR to each individual credit facility granted to an obligor. Risk ratings quantify the quality of individual facilities, credits, and portfolios. If RRs are accurately and consistently applied,

Page 271 then they provide a common understanding of risk levels and allow for active portfolio management. An RRS also provides the initial basis for capital charges used in various pricing models. It can also assist in establishing loan reserves. The RRS can be used to rate credit risks in most of the major corporate and commercial sectors, but it is unlikely to cover all business sectors. 7 The use of internal rating systems raises lots of issues. For example: what is the meaning of being in risk rating category X? Does it mean that the obligors in this category have an expected default probability (EDP) within a prespecified range? Or, is the rating associated with an expected loss given default? What is the horizon over which these estimations are derived? For instance, for the rating system to be consistent with the credit migration approach to modeling credit risk, each rating class should correspond to a range of default probabilities over a one-year period. The internal ratings approach has practical implications for supervisors. Some key considerations will have to be addressed when assessing a bank's rating system: is the number of gradations appropriate to distinguish among the range of risks? How can the bank link the rating to a measurable credit loss? Are all the appropriate risk factors incorporated? Notwithstanding these issues, the internal ratings approach is exciting because it would pave the way to the adoption of full credit risk modeling for the banking book in the future. The 1999 Basle consultative paper for a new capital adequacy framework (Basle, 1999) provides insight into the regulator's view of the role that an RRS can play in attributing regulatory capital. A typical RRS, as shown in Table 7.5, includes a category 0 to capture government debt (say, Canadian or U.S. federal government debt). Category 1 is reserved for the highest credit quality of corporate debt. The risk grades below A (e.g., BBB) are often split (say, into 4 and 5) to obtain greater resolution. The obligor rating represents the probability of default by a borrower in repaying its obligation in the normal course of business. The facility rating represents the expected loss of principal and/or interest on any business credit facility. It combines the likelihood of default by a borrower and the conditional severity of loss, should default occur, from the credit facilities available to that borrower.

Page 272 Table 7.5 Risk Rating Continuum (Prototype Risk Rating System) The steps in the RRS (nine, in our prototype system) typically start with a financial assessment of the borrower (initial obligor rating) which sets a floor on the obligor rating (OR). A series of further steps (four) arrive at a final obligor rating. Each one of Steps 2 to 5 may result in a downgrade of the initial rating attributed at Step 1. These steps include analyzing the managerial capability of the borrower (Step 2), examining the borrower's absolute and relative position within the industry (Step 3), reviewing the quality of the financial information (Step 4) and the country risk (Step 5). The process ensures that all credits are objectively rated using a consistent process to arrive at accurate ratings. Additional steps (four, in our example) are associated with arriving at a final facility rating, which may be above or below the final obligor rating. These steps include examining third-party support (Step 6), factoring in the maturity of the transaction (Step 7), reviewing how strongly the transaction is structured (Step 8), and

Page 273 assessing the amount of collateral (Step 9). The process, by steps, is described in detail in Sections 5 to 7 of this chapter. One needs to determine which entity (or group of entities) one is rating. For example, the analysis of a group credit involves calculating the obligor rating for the entire group of entities, provided that all the important entities and borrowers are cross-guaranteed. If this is not the case, then one should rate any such borrower individually. If there are businesses or companies in different industries, or with different financial characteristics, then one often focuses on either the dominant entity (if there is one) or a balance of the important components, with specific recognition of any weak links. A single entity might have a number of credit facilities with the bank that have different priority rules in case of bankruptcy. In this case, one must rate each facility with the credit. Conversely, if a number of facilities for a customer have similar characteristics (i.e., there are no distinguishing risk factors between the facilities), then one should apply the same facility rating to each facility. 3.2 Measuring Default Probabilities and Recovery Rates "How accurate are ratings?" asks Moody's in its Credit Ratings and Research (1995, p.5). The answer is provided in Figure 7.4, which shows the average cumulative default rates for corporate bond issuers for each rating category over bond holding periods of one year up to 20 years after bond issuance. The data are for the period 1970 to 1994. It can be seen that the lower the rating, the higher the cumulative default rates. The Aaa and Aa bonds experienced very low default rates, and after 10 years less than 1 percent of the issues had defaulted. Approximately 40 percent of the B-rated issues, however, had defaulted after 10 years. Figure 7.5 shows the average default rates within one year for different bond ratings during the period 1983 to 1993. In one year over 16 percent of the B-rated bonds defaulted, while the rate is 3 percent for the Ba3 bonds, and almost zero for the Aaa, Aa, and A bonds. Credit rating systems can also be compared to multivariate credit scoring systems to evaluate their ability to predict bank-

Page 274 Figure 7.4 Cumulative Default Rates for Corporate Bonds, 1970 1994 Source: Moody's Credit Ratings and Research, 1995. ruptcy rates and also to provide estimates of the severity of losses. Altman and Saunders (1998) provide a detailed survey of credit risk measurement approaches. They compare four methodologies for credit scoring: (1) the linear probability model, (2) the logit model, (3) the probit model, and (4) the discriminant analysis model. The logit model assumes that the default probability is logistically distributed, and applies a few accounting variables to predict the default probability. Martin (1977), West (1985), and Platt and Platt (1991) examine the logit model and find it useful in predicting bankruptcies. The linear probability model is based on a linear regression model, and makes use of a number of accounting variables to try to predict the probability of default. The multiple discriminant analysis (MDA), proposed and advocated by Altman

Page 275 Figure 7.5 One-Year Default Rates by Rating, 1983 1993 Source: Moody's Credit Ratings and Research, 1995. (see, for example, Altman, 1997) is based on finding a linear function of both accounting and marketbased variables that best discriminates between two groups: firms that actually defaulted, and firms that did not default. The linear models are based on empirical procedures: they search out the variables that seem best at predicting bankruptcies. They are not founded on a theory of the firm or on any theoretical stochastic processes for leveraged firms. Another shortcoming is that most models are based on accounting data that are updated at discrete points and thus do not fully convey the dynamics of the firms and the continuous process leading to bankruptcy. In Chapter 9 the economic approach to the estimation of bankruptcy is described. It is based on the contingent claim model whereby the equity of the firm is regarded as a call option on the assets of the firm. 4 Debt Rating and Migration Bankruptcy, whether defined as a legal or economic event, usually marks the end of a corporation in its current form. It is a discrete

Page 276 event, yet it is also the final point of a continuous process the moment when it is finally recognized that a firm cannot meet its financial obligations. Analysts that focus solely on the event of bankruptcy disregard a lot of useful information about the status of the firm, its total value, and the value of its liabilities. Of course, credit agencies do not focus simply on default. At discrete points in time they revise their credit rating of corporate bonds. This evolution of credit quality is very important for an investor holding a portfolio of corporate bonds. In a study published in November 1993, Moody's summarized its experience of rating 4700 long-term public debt issuers in the period May 1, 1923, to June 23, 1993. For the period 1950 to 1979, 4.44 percent of the companies changed their ratings within a year, with the proportion of upgraded companies (2.26 percent) slightly above that of downgraded companies (2.18 percent). For the period 1980 to 1993 the change of rating intensified to 10 percent, but the proportion of downgraded companies more than tripled to 6.82 percent of the rated companies. Table 7.6 provides data on upgrades and downgrades from 1983 through 1993, the period that followed the introduction of the numerical modifiers to the letter rating in 1982. This period is characterized by deteriorating credit quality. The percentage of downgrades is substantially higher than the percentage of upgrades. The last column summarizes the drift of credit quality by counting the total number of numerical notches changed for upgrades minus the total number changed for downgrades, divided by the number of rated companies. The "Rating Activity" column and the "Drift" column take into consideration the size of the change in rating and not only the event of rating change. 8 Actually, 57 percent of all rating changes were of one notch only, 30 percent of two notches, and 7 percent of three notches. These changes are for the numerical modifiers to the letter ratings. One letter change, for example from Baa to Ba, occurred in 89 percent of the cases of letter change, and in 9 percent of the cases the change was two letters. Using transition matrices, we can see how different rating categories have changed through time. Table 7.7 is based on Moody's experience from 1970 to 1993, and it contains the empirical results for the migration from one credit risk category to all other credit

Page 277 TABLE 7.6 Long-Term, Modified Rating Changes by Year, 1983 1993 Upgraded Issuers Downgraded Issuers Number Percentage Number Percentage Rating Activity % Drift % 1983 122 8.91 148 10.81 32.85 4.60 1984 191 12.46 173 11.29 42.80 3.98 1985 169 9.37 237 13.14 47.17 18.48 1986 171 8.02 345 16.19 50.40 24.98 1987 159 6.22 274 10.72 35.87 10.79 1988 178 6.00 324 11.04 38.97 11.82 1989 168 5.12 337 10.37 32.97 16.51 1990 138 3.82 489 13.52 33.88 21.21 1991 153 3.99 485 12.65 29.26 16.38 1992 178 4.33 451 10.98 25.27 11.54 1993 1 238 5.40 450 10.21 23.88 8.53 1 The numbers for 1993 are assimilated from data available from January 1, 1993, through June 22, 1993.

Page 278 TABLE 7.7 Transition Matrices for Bond Ratings for 1, 2, 5, and 10 Years Part A: One-Year rating transition matrix To Aaa % Aa % A % Baa % Ba % B % Caa % Default % WR % From Aaa 89.6 7.2 0.7 0.0 0.0 0.0 0.0 0.0 2.5 Aa 1.1 88.8 8.9 0.3 0.2 0.0 0.0 0.0 2.8 A 0.1 2.5 89.0 5.2 0.6 0.2 0.0 0.0 2.5 Baa 0.0 0.2 5.2 85.3 5.3 0.8 0.1 0.1 3.0 Ba 0.0 0.1 0.4 4.7 80.1 6.9 0.4 1.5 5.8 B 0.0 0.1 0.1 0.5 5.5 75.7 2.0 8.2 7.8 Caa 0.0 0.4 0.4 0.8 2.3 5.4 82.1 20.3 8.4 Part B: Two-Year rating transition matrix To Aaa % Aa % A % Baa % Ba % B % Caa % Default % WR % From Aaa 80.9 12.6 1.6 0.1 0.1 0.0 0.0 0.0 4.8 Aa 2.2 78.6 12.1 1.1 0.8 0.0 0.0 0.1 5.4 A 0.1 4.9 79.6 8.6 1.5 0.5 0.1 0.1 4.6 Baa 0.1 0.5 9.8 73.3 8.6 1.6 0.2 0.4 5.6 Ba 0.1 0.1 0.8 8.4 64.4 10.5 0.7 4.3 10.7 B 0.0 0.2 0.2 1.0 8.2 58.8 2.4 14.7 14.6 Caa 0.0 0.4 0.4 2.2 3.1 8.7 44.5 27.1 13.5 (Continued) risk categories within one year, two years, five years, and 10 years. The values on the diagonals of the transition matrix show the percentage of bonds that remained in the same risk category at the end of the specified time period as they occupied at the beginning of the specified time period. For example, from Part A of the table, we see that 89.6 percent of the bonds rated Aaa stayed in the same rating category a year

Page 279 (Continued) TABLE 7.7 Transition Matrices for Bond Ratings for 1, 2, 5, and 10 Years Part C: Five-year rating transition matrix To Aaa % Aa % A % Baa % Ba % B % Caa % Default % WR % From Aaa 62.5 21.8 4.9 0.5 0.7 0.2 0.1 0.2 9.1 Aa 5.5 52.9 22.3 3.9 1.8 0.5 0.0 0.4 12.7 A 0.7 9.9 59.6 15.0 3.9 1.1 0.2 0.8 9.3 Baa 0.2 1.9 18.8 49.7 12.6 3.2 0.3 1.7 11.6 Ba 0.2 0.5 3.6 13.6 37.4 12.8 0.8 10.1 21.2 B 0.1 0.1 0.7 3.1 10.3 31.8 1.7 24.8 27.4 Caa 0.0 0.0 0.6 7.8 5.8 14.0 19.9 35.1 17.0 Part D: Ten-year rating transition matrix To Aaa % Aa % A % Baa % Ba % B % Caa % Default % WR % From Aaa 47.1 31.5 8.8 3.6 1.7 0.2 0.1 1.0 6.0 Aa 8.4 33.6 30.6 9.6 3.3 0.8 0.2 1.3 12.1 A 0.8 14.8 43.0 17.9 5.9 2.5 0.4 1.1 13.9 Baa 0.3 4.7 28.4 29.9 13.2 4.2 0.4 4.0 17.0 Ba 0.4 1.7 10.0 18.6 19.8 10.4 0.6 13.9 24.6 B 0.8 0.0 4.9 6.1 11.6 16.5 1.4 30.2 28.5 Caa 0.0 0.7 4.3 14.6 6.8 8.5 8.5 48.7 8.5 Source: Carty and Fons (1993). later. Observe that 7.2 percent were downgraded to Aa, 0.7 percent downgraded to A, etc. A firm rated Baa stayed in the same risk category after two years in 73.3 percent of the cases (see Part B), while there was a 9.8 percent chance of the firm being upgraded to a rating of A. Bonds rated Baa had a 0.4 percent chance of defaulting within two years. The last column, ''WR," reports the percentage of issuers that had their ratings withdrawn at the end of the period.

Page 280 It is interesting to note that bonds with an initial rating of Caa defaulted in 27.1 percent of the cases within two years, and that 35.1 percent of them defaulted after five years. For bonds rated Aaa the percentages were 0.0 percent and 0.2 percent for two years and five years, respectively. After five years, only 62.5 percent of the Aaa-rated bonds had maintained their initial rating, and about 28 percent of the Aaa bonds were downgraded, while over 9 percent had their ratings withdrawn. Issuers rated Aaa can either maintain their rating or be downgraded. Caa-rated bonds can maintain their rating, be upgraded, or go into default. But what of Baa-rated bonds? Based on their history, they seem to have an equal chance of being upgraded or downgraded within a period of one and two years. However, over periods of five and 10 years they seem more likely to be upgraded than downgraded. The transition matrices play a major role in the credit evaluation system of JP Morgan's CreditMetrics described in Chapter 8. Moody's also supplies transition matrices for the modified rating categories, i.e., categories with number modifiers (e.g., A2) added to the letter ratings. The number modifiers, as pointed out in Section 2.2, enable Moody's to further differentiate the letter rating from, say, the highest quality A-rated credit (i.e., A1) to the lowest A-rated credit quality (i.e., A3) with a mid-range allowance for credit quality (i.e., A2). Additional statistics are given for issuers of short-term instruments. Moody's also suggests that a Wiebull distribution most closely models the characteristics of bond ratings over their life spans. Figures 7.6 and 7.7 provide, respectively, the estimated average length of letter rating lives and the average length of modified rating lives. Based on past transition experience, researchers suggest various methodologies to estimate transition probabilities. Altman and Kao (1992a, 1992b) use the Markovian stable and unstable models. Bennet (1987) analyzed the rating migration of bank's assets. In a recent article, Altman (1998) compares expected rating changes for Moody's and S&P over the period 1970 to 1996. The two agencies include in their statistics both newly issued bonds as well as seasoned bonds of all ages at a given date. They follow the migration for each pool of bonds for up to 15 years after the initial period. The major problem with this analysis is that while all the bonds in

Page 281 Figure 7.6 Average Length of Letter Rating Lives Source: Moody's Credit Ratings and Research, 1995. the pool initially had the same credit rating, they had different maturities. Older bonds have a greater tendency to migrate than newly issued bonds. Hence the pools may contain biases. Altman and Kao (1992) investigate the migration of ratings from the initial bond rating until up to 10 years later. Figure 7.7 Average Length of Modified Rating Lives Source: Moody's Credit Ratings and Research, 1995.

Page 282 Table 7.8 is reproduced from Altman (1998). It shows the one-year transition matrix for long-term senior bonds based on statistics of Moody's, S&P, and Altman and Kao (A/K). 9 The time period covered by the different studies is not identical; this explains some of the differences, since migration is time-dependent, and is probably affected by macroeconomic trends. The aging problem affects the results, and consistently, the values on the diagonal for A/K are higher than for Moody's and S&P. In A/K the bonds in each initial category are newly issued and therefore have longer maturities. A/K also adjust for rating withdrawn (RW) since in many cases the fact that a bond is no longer rated is due to mergers and acquisitions of the issuer, and hence to early redemption of the principal. 5 Financial Assessment (Step 1) 5.1 Introduction This step formalizes the thinking process associated with a good credit analyst (or good equity analyst) whose goal is to ascertain the financial health of an institution. For example, the credit analyst would study the financial reports to determine if the earnings and cash flows are sufficient to cover the debt. The credit analyst will study the degree to which the trends associated with these "financials" are stable and positive. The credit analyst would also want to analyze the degree to which the assets are of high quality, and make sure that the obligor has substantial cash reserves (e.g., substantial working capital 10 ). The analyst would also want to examine the firm's leverage. Similarly, the credit analyst would also want to analyze the degree to which the firm had access to the capital markets, and whether it has an appropriate capacity to borrow money. The rating should reflect the financial position and performance of the company and its ability to withstand possibly unexpected financial setbacks. This is a key step in the credit assessment. 5.2 Procedure The obligor will almost always be the borrower (or group of borrowers). Nevertheless, a guarantor, in certain circumstances

Page 283 TABLE 7.8 Rating Transition Matrix One-Year Horizon Aaa/AAA Aa/AA A/A Baa/BBB Ba/BB B/B Caa/CCC Def C/D RW AAA (A/K) 94.3 5.5 0.1 0.0 0.0 0.0 0.0 0.0 Aaa (M) 88.3 6.2 1.0 0.2 0.0 0.0 0.0 0.0 4.3 AAA (S&P) 88.5 8.1 0.7 0.1 0.1 0.0 0.0 0.0 2.6 AA (A/K) 0.7 92.6 6.4 0.2 0.1 0.1 0.0 0.0 Aa (M) 1.2 86.8 5.8 0.7 0.2 0.0 0.0 0.0 5.4 AA (S&P) 0.6 88.5 7.6 0.6 0.1 0.1 0.0 0.0 2.4 A (A/K) 0.0 2.6 92.1 4.7 0.0 0.2 0.0 0.0 A (M) 0.7 2.3 86.1 4.7 0.6 0.1 0.0 0.0 6.0 A (S&P) 0.1 2.3 87.6 5.0 0.7 0.2 0.0 0.4 3.6 BBB (A/K) 0.0 0.0 5.5 90.0 2.8 1.0 0.1 0.3 Baa (M) 0.0 0.3 3.9 82.5 4.7 0.6 0.1 0.3 7.7 BBB (S&P) 0.0 0.3 5.5 82.5 4.7 1.0 0.1 0.2 5.7 BB (A/K) 0.0 0.0 0.0 6.8 86.1 6.3 0.9 0.0 Ba (M) 0.0 0.1 0.4 4.6 79.0 5.0 0.4 1.1 9.4 BB (S&P) 0.0 0.1 0.6 7.0 73.8 7.6 0.9 1.0 8.9 B (A/K) 0.0 0.0 0.2 1.6 1.7 93.7 1.7 1.1 B (M) 0.0 0.0 0.1 0.6 5.8 73.1 3.5 10.5 7.8 B (S&P) 0.0 0.1 0.2 0.4 6.0 72.8 3.4 4.9 12.2 CCC (A/K) 0.0 0.0 0.0 0.0 0.0 2.8 92.5 4.6 Caa (M) 0.0 0.0 0.0 0.3 1.3 5.3 71.9 12.4 8.8 CCC (S&P) 0.2 0.0 0.3 1.0 2.2 9.6 53.1 19.3 14.2 (All numbers are percent.) Sources and Key: A/K: Altman and Kao (1971 1989) from Altman and Kao (1992A,b) Newly issued bonds. M: Moody's (1920 1996) from Moody's (1997) Cohorts of bonds. S&P: Standard & Poor's (1981 1996) from Standard & Poor's (1997) Static pools of bonds. RW: rating withdrawn. Source: Altman (1998).

Page 284 (outlined below) may be substituted and regarded as the obligor. For example, one may substitute a guarantor for the borrower where the credit risk lies solely on the guarantor (i.e., the borrower's position is not a meaningful factor) and the guarantor is a large national (or international) entity warranting, say, an investment-grade rating (i.e., a risk rating (RR) of 4 or better). Further, the debt needs to be structured so as to ensure that the bank will not be in an inferior position to other obligations of the guarantor, and the bank must make sure that a "clean 100 percent" guarantee is held. 11 One needs to monitor the guarantor's performance with the same care as if the guarantor were the direct borrower. A prototype financial assessment table encompassing the risk ratings 4 is shown in Table 7.9. The three main assessment areas, as illustrated at the top of Table 7.9, are: (1) earnings (E) and cash flow (CF); (2) asset values (AV), liquidity (LIQ), and leverage (LEV); and (3) financial size (FS), flexibility (F), and debt capacity (DC). A measure for earnings/cash flow in column 1 would include interest coverage such as EBIT/interest expense and EBITDA/interest expense. 12 A measure for leverage in column 2 would include the current ratio, which is defined as current assets divided by current liabilities. A measure for leverage in column 2 would also include debt-to-net-worth ratios such as total liability/equity or (total liabilities-short-term debt)/equity. One would calculate an RR for each of the three assessment areas and then arrive at an assessment of the best overall risk rating. 13 This is the initial obligor rating (OR). The remaining portions of a prototype financial assessment table for RR 4 are shown in Table 7.9. There will be cases and/or industries where one of the three main assessment areas should be more heavily (or lightly) weighted when arriving at the overall financial assessment. The use of judgment is essential. One should benchmark any assessment to those of other companies in the same industry grouping. One needs to emphasize the current year's performance, with some recognition of the previous few years as appropriate when assessing the earnings and cash flow category. Cash flow is assessed using whatever methodology is most appropriate to the industry or individual situation (e.g., EBITDA). When assessing companies

Page 285 TABLE 7.9 Step 1 Financial Assessment Asset Values (AV) Financial Size (FS) Earnings (E) Liquidity (LIQ) Flexibility (F) RR Cash Flow (CF) Leverage (LEV) Debt Capacity (DC) 4 Very satisfactory earnings and cash flow with substantial extra coverage Positive and quite consistent/stable trends Assets of above average quality Good liquidity/working capital Better than average leverage Appropriate matching of tenor of liabilities to assets General access (rated BBB+/BBB) to capital markets; may experience some barriers due to difficult market or economic conditions Ready access to alternate financing through banks or other financial institutions, if sought Bank debt modest with large unused capacity in cyclical industries one should adjust the financial results and key ratios so that the cyclical effect is incorporated. This is reasonable so long as downturns are within the scope of a normal cycle (i.e., not a remote fundamental correction). This means that strong performance during a very positive economic period should be modified downward somewhat (and vice versa during a weak period). When assessing the financial size, flexibility, and debt capacity category, the size of market capitalization will also be an important factor. "Access to capital markets" in this third assessment area refers to the demonstrated ability (or potential in the near-term) to issue public securities (equities or medium- to long-term debt instruments), which generally will have necessitated the assignment of a public rating. For private or smaller companies one should consider the ability to access these markets. If financial information/data are not available (such as for new ventures, projects, etc.) then "proforma" data are often acceptable. 5.3 Industry Benchmarks The analysis of the competitive position and operating environment of a firm helps in assessing its general business risk profile.

Page 286 This leads to the calibration of the quantitative information drawn from the financial ratios for the firm, using industry benchmarks. The ratios summarize information on the profitability and interest coverage of the issuer, on its capital structure (i.e., leverage), asset protection, and cash flow adequacy. The major ratios considered include those stipulated in Box 7.1. Appendix 1 provides a detailed definition of each of the key ratios. Table 7.10 shows the interaction between the general business risk 14 assessment of a company and two selected financial ratios (ratios 3 and 8 in Box 7.1) in determining the rating categories. A company with an excellent business can assume more debt than a company with average business possibilities. For example, a company with an excellent business position will be able to take on a debtto-total-capitalization ratio (ratio 8 in Box 7.1) of 50 percent, and qualify for rating category A. By contrast a company with average business possibilities will be able to take on a debt-to-totalcapitalization ratio of only 30 percent if it is to qualify for rating category A. Table 7.11 provides data on average ratios for risk categories for three overlapping periods (1992 to 1994, 1993 to 1995, 1994 to 1996). The table indicates that the ordinal nature of the categories corresponds well, on average, to the financial ratios. For example, if we examine the EBIT interest coverage ratio (i.e., EBIT divided by interest expense), then we would observe that the median for the AA credit class for the 1994 to 1996 period was 11.06, while the BB was 2.27. The ratio for the AA credit class ranged from a low BOX 7.1 1. EBIT interest coverage ( ) 2. EBITDA interest coverage ( ) 3. Funds from operations/total debt (%) 4. Free operating cash flow/total debt (%) 5. Pretax return on capital (%) 6. Operating income/sales (%) 7. Long-term debt/capital (%) 8. Total debt/capitalization (%)

Page 287 TABLE 7.10 Guidelines for Adjustments in Two Financial Ratios As a Function of the Business Risk Profile to Qualify to a Given Rating Category Funds From Operations/Total Debt Guidelines (%) Rating Category Company Business Profile AAA AA A BBB BB Excellent business position 80 60 40 25 10 Above average 150 80 50 30 15 Average 105 60 35 20 Below average 85 40 25 Vulnerable 65 45 Total Debt/Capitalization Guidelines (%) Rating Category Company Business Profile AAA AA A BBB BB Excellent business position 30 40 50 60 70 Above average 20 25 40 50 60 Average 15 30 40 55 Below average 25 35 45 Vulnerable 25 35 Source: S&P Corporate Ratings Criteria, 1998. of 11.06 to a high of 9.67 over the three (1992 to 1994, 1993 to 1995, 1994 to 1996) three-year overlapping sample periods, while the ratio for the BB class ranged from 2.09 to 2.27. 5.4 Combining Balance Sheet, Income Statement, and Ratio Analyses The analysis of loans for the purpose of arriving at a risk rating requires one to think through certain classic relationships between balance sheet, income statement, and ratio analysis. We will first examine a few of these relationships for purely illustrative purposes and then show how they might be useful in arriving at a risk rating.

Page 288 TABLE 7.11 Key Industrial Financial Ratios for Rating Categories U.S. Industrial Long-Term Debt Three-Year (1994 to 1996) Medians AAA AA A BBB BB B 1. EBIT interest coverage ( ) 16.0 11.0 6.2 4.1 2.2 1.1 2. EBITDA interest coverage ( ) 20.3 14.9 8.5 6.0 3.6 2.2 3. Funds from operations/total debt (%) 116.4 72.3 47.5 34.7 18.4 10.9 4. Free operating cash flow/total debt (%) 76.8 30.5 18.8 8.4 2.4 1.2 5. Pretax return on capital (%) 31.5 23.6 19.5 15.1 11.9 9.1 6. Operating income/sales (%) 24.0 19.2 16.1 15.4 15.1 12.6 7. Long-term debt/capital (%) 13.4 21.9 32.7 43.4 53.9 65.9 8. Total debt/capitalization (%) 23.6 29.7 38.7 46.8 55.8 68.9 U.S. Industrial Long-Term Debt Three-Year (1993 to 1995) Medians AAA AA A BBB BB B 1. EBIT interest coverage ( ) 13.5 9.6 5.7 3.9 2.1 1.1 2. EBITDA interest coverage ( ) 17.0 12.8 8.1 6.0 3.4 2.1 3. Funds from operations/total debt (%) 98.2 69.1 45.5 33.3 17.7 12.8 4. Free operating cash flow/total debt (%) 60.0 26.8 20.9 7.2 1.4 (0.9) 5. Pretax return on capital (%) 29.3 21.4 19.1 13.9 12.0 9.0 6. Operating income/sales (%) 22.6 17.8 15.7 13.5 13.5 12.3 7. Long-term debt/capital (%) 13.3 21.1 31.6 42.7 55.6 65.5 8. Total debt/capitalization (%) 25.9 33.6 39.7 47.8 59.4 69.5 U.S. Industrial Long-Term Debt Three-Year (1992 to 1994) Medians AAA AA A BBB BB B 1. EBIT interest coverage ( ) 18.0 9.7 5.3 2.9 2.1 1.0 2. EBITDA interest coverage ( ) 22.6 12.8 8.0 4.8 3.5 1.9 3. Funds from operations/total debt (%) 97.5 68.5 43.8 29.9 17.1 9.9 4. Free operating cash flow/total debt (%) 51.0 29.7 20.2 6.2 3.4 1.1 5. Pretax return on capital (%) 28.2 20.6 16.7 12.7 11.6 8.3 6. Operating income/sales (%) 22.0 17.7 15.2 13.2 13.6 11.6 7. Long-term debt/capital (%) 13.2 19.7 33.2 44.8 54.7 65.9 8. Total debt/capitalization (%) 25.4 32.4 39.7 49.5 60.1 73.4 EBIT refers to earnings before interest and taxes. EBITDA refers to earnings before interest, taxes, depreciation, and amortization. Source: S&P Corporate Ratings Criteria, 1998.

Page 289 BOX 7.2 BALANCE SHEET Assets Liabilities Key Relationships CA CL WC = CA - CL FA LTD FW = FA - LTD NW NW = WC + FW TA = CA + FA TL = CL + LTD TA = TL + NW Total assets (TA), as shown in Box 7.2, are identically equal to total liabilities (TL) and net worth (NW). Current assets (CA) are identical to current liabilities (CL) and working capital (WC): 1. TA = TL + NW 2. WC = CA CL Total assets are also composed of current assets (CA) and fixed assets (FA), which is: 3. TA = CA + FA Total liabilities are composed of current liabilities (CL) plus long-term debt (LTD), as follows: 4. TL = CL + LTD If we refer to LTD + NW as permanent capital, then by rearranging our terms the ''working capital" can be shown to equal the permanent capital minus the fixed assets: 5. WC = LTD 15 + NW FA 16 Fixed worth (FW) is defined as fixed assets long term debt: 6. FW = FA LTD Net worth can be expressed as working capital plus fixed worth. 7. NW = WC + FW A working capital leverage ratio would express the riskiness of the current capital structure. One would also analyze certain key

Page 290 ratios. For example, a ratio of current liabilities to working capital (called the working capital leverage ratio) is analogous to the leverage ratio of total liabilities to net worth. 8. Working capital leverage ratio = CL/WC. The leverage ratio expresses the riskiness of the overall capital structure, or how long-term debt is supported by equity: 9. Leverage ratio = TL/NW. 10. Current ratio = CA/CL. A prototype high-level customer financial information (CFI) report is shown in Table 7.12 for General Motors Acceptance Corporation. Such a report is typically produced to facilitate credit analysis (at, say, the senior credit committee meeting of the bank). The CFI report is divided into a balance sheet, income statement, and ratio analysis section. The ratio analysis section is further subdivided into leverage ratio and solvency ratio. An experienced credit analyst can quickly analyze such a report and get a "feel" for the financial assessment portion of the risk rating process. For example, one may analyze the leverage ratio (say, total liabilities/equity), solvency ratio (say, interest coverage), or other key financial analysis measures (see Appendix 2) as part of arriving at the appropriate financial assessment. 6 First Group of Adjustment Factors for Obligor Credit Rating 6.1 Management and Other Qualitative Factors (Step 2) This second step considers the impact on an obligor rating of a variety of qualitative factors such as discovering unfavorable aspects of a borrower's management. We will assume for illustrative purposes that this Step 2 analysis has no effect on the RR if the obligor seems to reach an acceptable standard, but that it may bring about a downgrade if standards are not acceptable. A typical Step 2 approach would require one to examine day-to-day account operations (AO) and assess management (AM), as well as perform an environmental assessment (EA) and examine contingent liabilities (CL), etc.

Page 291 TABLE 7.12 Example Customer Financial Information Report: Balance Sheet, Income Statement, and Ratio Data General Motors Acceptance Corporation Factors 12/31/1997 12/31/1996 Current assets (CA) 44,658 41,598 Current liabilities (CL) 64,288 50,469 Balance Sheet Working capital (WC=CA CL) 19,630 8,871 Fixed assets (FA) 64,661 56,980 Mortgages/other (LTD) 36,275 39,841 Fixed worth (FW=FA-LTD) 28,386 17,139 Net worth (NW=WC+FW) 8,756 8,268 Sales for year 16,595 15,974 Operating profit (EBIT) 7,471 7,415 Depreciation and amortization (DA) 4,735 4,668 Bad debts 523 669 Income Statement Income taxes 913 837 Net profit/loss 1,301 1,241 Dividends/drawings 750 1,200 Sundry adjustments 63 42 Net capital expenses 0 0 Interest expense (I) 5,256 4,938 Leverage Ratios: Total liabilities/equity 11.49 1 10.92 (Total liab sub-debt)/equity 44.49 10.92 2 Ratios Working capital 0.69 3 0.82 Solvency Ratios: Interest coverage (EBIT/I) 1.42 4 1.42 Cash interest coverage (EBITDA/I) 2.32 5 2.37 1. TL = CL + LTD = 64,288 + 36,275 = 109,124; TL Equity (NW) = 109, 124/8, 756 = 11.49. 2. No subordinated debt in 1996. 3. Working capital current ratio = CA/CL = 44,658/64,288 = 0.69. 4. EBIT = operating profit. Note that EBIT/I = 7471/5256 = 1.42. 5. EBITDA = EBIT + DA = 7471 + 4735 12,206. Note that EBITDA/I = 12,206/5256 = 2.32. If one is examining the day-to-day AO, then one would ask a series of carefully structured questions. For example, if the financial and security reporting is on a timely basis, is it of good quality? Does it satisfactorily explain significant variations from projections? One would also ask if the credit limits and terms are

respected and examine whether any past requests for temporary excesses, terms, etc., were made before rather than after the fact. One would also ask if the company honors its obligations with creditors (legitimate disputes aside), as evidenced by a lack of writs, lawsuits, judgments, etc. Page 292 One would ask, in terms of performing a management assessment, if management skills are sufficient for the size and scope of the business. This would include examining if management has a satisfactory record of success as well as appropriate industry experience. One should also examine if management has adequate "depth"; for example, are succession plans in place? One would ask a series of practical questions. For example: is there an informed approach to identifying, accepting, and managing risks? Does management stay current on how to conduct business operations, introducing and updating methods and technology when warranted? Does management address problems promptly, exhibiting the will to take hard decisions as necessary and with an appropriate balance of shortto long-term concerns? Is a reasonable business and financial plan in place, which does not depend on unrealistic levels of business growth or profitability improvement? Is management remuneration (cost to firm) prudent and appropriate to the size and financial strength/progress of the company? One should ask from an EA point of view if management is aware of, monitors and complies with all relevant environmental regulations and practices. One should also examine any contingent liabilities, e.g., litigation, or warranty claims. 6.2 Industry Ratings Summary (Step 3A) This portion of the third step recognizes the very important effect of an industry rating based on the type of industry and the relative position of the borrower (i.e., their tier assessment) within their industry. Experience has shown that poorer-tier performers in weak, vulnerable industries are major contributors to credit losses. To do this, the analyst needs to rate each industry type on, say, a scale of 1 to 5 (see Table 7.13). One should provide an industry assessment (IA) ratings scheme for each industry broken down into selective sub-industry groupings. For example, the forest products industry may be broken down into a sub-industry

Page 293 grouping such as wood products. Similarly, the mining industry may be broken down into sub-industry groupings such as gold mines, base metal mines, etc. A rating is assigned to each of the industry groupings. To calculate the industry assessment, the analyst first assigns a score of, say, 1 (minimal risk) to 5 (very high risk) for each of a set of, say, eight criteria established by the bank. For example, one can describe the industry rating in terms of competitiveness (see Table 7.13 for detailed definitions), trade environment, regulatory framework, restructuring, technological change, financial performance, long-term trends affecting demand, and vulnerability to macroeconomic environment. The sum of the scores, which will range from 8 (most favorable) to 40 (least favorable), can then be converted to an industry rating. For example, the asset would be rated 1 if it has a score ranging from 8 to 11. Similarly, a total score of between 9 and 19 yields an industry score of 2; between 20 and 27 a score of 3; between 27 and 35 a score of 4; and a score of 5 for a total score of between 36 to 40. TABLE 7.13 Rating the Competitiveness of an Industry Industry Risk Minimal 1 Low 2 Medium 3 High 4 Very High 5 Competitiveness The potentital of the industry to sell in its domestic market and/or external markets based only on: cost structure (determined by factors such as economies of scale, capital intensity, input costs, location, infrastructure, and use of appropriate technology), international reputation, and effectiveness in targeting market niches. On balance, the combination of the relevant listed factors makes the industry very competitive. On balance, the combination of the relevant listed factors makes the industry somewhat competitive. The relevant listed factors have off-setting impacts on the competitiveness of the industry. On balance, the combination of the relevant listed factors makes the industry somewhat uncompetitive. On balance, the combination of the relevant listed factors makes the industry very uncompetitive.

Page 294 Competitiveness (Table 7.13) can be defined as the potential of the industry to sell its products in its domestic market and/or external markets, given its cost structure (determined by factors such as economies of scale, capital intensity, input costs, location, infrastructure, and use of appropriate technology), international reputation, and effectiveness in targeting market niches. The trade environment can be defined as all the institutional factors that affect interjurisdictional commerce in goods and services, including trade agreements that have an impact (or potential impact) on the industry. The regulatory framework can be defined as the legal/institutional setting including laws and regulations of applicable levels of government direct and indirect taxation, grant programs, trade finance, and subsidies. One needs to take into account present policies and trends, the industry's ability to absorb and influence these policies and trends, and the impact of both supply and demand. Restructuring can be defined as the impact of the process of adjusting (often through a reduction in capacity or employees) to a change in market conditions, such as demand patterns, technology, number and quality of competitors, or regulations. Technological change can be defined as industry vulnerability to technological change that could result in changing costs, an alteration in the range of products or services of the industry, or an alteration in the range/price of competitive products/services. Knowledge of previous technological change and current relevant global research and development efforts must be taken into account. Financial performance can be defined as an assessment based on the present level, trends, and sustainability of standard ratios such as return on equity, interest coverage, current ratio, debt/equity, and debt/cash flow. Long-term trends that affect demand include demographics (i.e., age structure, gender distribution, composition, and wealth distribution of the relevant market), vintage of durables and infrastructure (age of fleet and age and condition of roads, bridges, etc.), and lifestyle changes and consumer attitudes. Vulnerability to macroeconomic environment describes how sensitive the industry is to economic downturns, fiscal policy, movements in interest rates and exchange rates, and other macroeconomic variables.

Page 295 Appendix 3 of this chapter offers a condensed example of an illustrative prototype assessment of the telecommunication industry (Appendix 3A) as well as the footwear and clothing industry (Appendix 3B). 6.3 Tier Assessment (Step 3B) The second part of Step 3 involves establishing tier assessment (TA) the relative position of each business within its own industry. This is an important survival factor, particularly during downturns. One can use the criteria and process used to assess industry risk to determine a company's relative position in one of relative tiers say, on a scale of 1 to 4 within an industry. A business should be ranked against its relative competition. For example, if the company supplies a product/service that is subject to global competition, then it should be ranked on a global basis. If the company's competitors are by nature local or regional, as are many retail businesses, then it should be ranked on that basis, while recognizing that competition may increase. If a business is local but has no local competitors, e.g., a local cable operator, then it should be ranked against such companies in other areas, with some recognition of the benefit of the exclusivity of its market (assuming that this is likely to continue). Tier 1 players are major players with a dominant share of the relevant market (local, regional, domestic, international, or niche). They have a diversified and growing customer base with low production costs that are based on sustainable factors (such as a diversified supplier base, economies of scale, location and resource availability, continuous upgrading of technology, etc.). Such companies respond quickly and effectively to changes in the regulatory framework, trading environment, technology, demand patterns, and macroeconomic environment. Tier 2 players are important or above-average industry players with a meaningful share of the relevant market (local, regional, domestic, international, or niche). Tier 3 players are average (or modestly below average) industry players, with a moderate share of the relevant market (local, regional, domestic, international, or niche). Tier 4 players are weak industry players and have a declining customer base. They have a high cost of production due to factors such as low leverage with suppliers, obsolete technologies, etc.

Page 296 6.4 Industry/Tier Position (Step 3C) This is the final part of the third step (Step 3C). If one can combine assessments of the health of the industry (i.e., industry rating) and the position of a business within its industry, then one can assess the vulnerability of any company (particularly during recessions). Low-quartile competitors within an industry class almost always have higher risk (modified by the relative health of the industry). One needs to combine the industry rating and the tier assessment using the grid in Table 7.14 to determine the "best possible" OR. The rating is "best possible" in the sense that it acts as a cap on the OR. While the rating can be lowered if the industry/tier assessment is weak, it will not be raised if it is strong. For example, if the industry rating assessment indicates that the industry rating is 2, and is considered to be tier 3, then the "best possible" obligor rating is 5. If Steps 1 and 2 had suggested a rating of 4, then Step 3 would require that this rating be lowered to 5. 6.5 Financial Statement Quality (Step 4) This fourth step recognizes the importance of the quality of the financial information provided to the analyst. Again, this step is not used to improve the rating, but to define the best possible OR. TABLE 7.14 Best Possible Obligor Rating (Given Initial Industry and Tier Ratings) Industry Rating (From Step 3A) 1 2 3 4 5 Tier assessment within industry (from Step 3B) Tier 1 Tier 2 Tier 3 No effect on rating Specific adjustments are provided with each row column combination Tier 4 9

Page 297 The bank must always be fully satisfied as to the quality, adequacy, and reliability of the financial statement information irrespective of the RR. This includes consideration of the size and capabilities of the accounting firm, compared to the size and complexities of the borrower and its financial statements. Exceptions may be made. For example, they may be appropriate in the case of subsidiaries of large international/national corporations where the obligor's financial statements are eventually consolidated into audited financial statements of the parent. One may also make exceptions for new entities (or certain specialized industries) as well as obligors in countries where accepted practices differ from North American standards. 6.6 Country Risk (Step 5) This fifth step adjusts for the effect of any country risk. Country risk is the risk that a counterparty, or obligor, will not be able to pay its obligations because of cross-border restrictions on the convertibility or availability of a given currency. It is also an assessment of the political and economic risk of a country. The economics department of a bank is typically involved in analyzing the macro- and micro-economic factors that allow an analyst to calculate a country risk rating. (Naturally, if the counterparty has all or most of its cash flow and assets in the local market, then one may skip this step.) A table should be developed to determine whether a country rating will affect the OR. Country risk exists when more than a prescribed percentage (say 25 percent) of the obligor's (gross) cash flow or assets is located outside of the local market. Country risk may be mitigated by hard dollar cash flow received/earned by the counterparty. Hard dollar cash flow refers to revenue in a major (readily exchangeable) international currency (primarily U.S. dollars, U.K. pounds, Euros, and Japanese yen, as well as Canadian dollars). If the obligor is strong, then short-term country risks (primarily trade finance and trading products) may warrant a better rating than the country. One may also mitigate country risk or improve the rating in a later step in the process. Obtaining political risk insurance (or similar mitigants) may also (partially) mitigate country risk.

Page 298 TABLE 7.15 Adjustment for Country Risk Division Country Ratings Excellent, very good, good, or satisfactory Adjustment to Obligor Rating None Fair Best possible obligor rating is 5 Selectively acceptable Best possible obligor rating is 6 Marginal/deteriorating Best possible obligor rating is 7 Again, Step 5 acts to limit the best possible rating. For example, if the client's operation has a country rating in the "fair" category, then the best possible OR is 5 (see Table 7.15). On the other hand, if the country is rated "selectively acceptable," then the best possible obligor rating is 6. A condensed version of an illustrative prototype country analysis is provided in Appendix 4. 7 Second Group of Adjustment Factors for Facility Rating 7.1 Third-Party Support (Step 6) This sixth step adjusts a facility rating (FR) where important third-party support is held. (This step can therefore be skipped if the guarantor was substituted for the borrower at the outset.) Considerable care and caution is necessary if ratings are to be improved because of the presence of a guarantor. In all cases, one must be convinced that the third party/owner is committed to ongoing support of the obligor. Typically, one establishes very specific rules for third-party support as described in Box 7.3. Based on the quality of the third-party support, the risk rating of the firm can be upgraded or downgraded. Personal guarantors and other undertakings from individuals, and guarantees for less than 100 percent of the indebtedness, do not qualify for consideration in this category.

Page 299 BOX 7.3 THIRD-PARTY SUPPORT Type of Support Guarantee Completion guarantee Keepwell agreement or operating agreement Comfort letter or ownership A 100 percent clean guarantee is held.* A 100 percent clean guarantee is held until completion of the project. A strong keepwell 17 or operating agreement is held and is considered legally enforceable. A comfort letter 18 is held or no written assurance is held. * See Note 11 for a definition of ''clean guarantee." 7.2 Term (Step 7) This seventh step recognizes the increased risk associated with longer-term facilities and the lower risk of very short term facilities. A standard approach is to combine the adjusted facility rating (after any third-party support adjustment, in Step 6) with the remaining term to maturity in order to determine the adjustment to the facility rating, as shown in the matrix in Table 7.16. One would also need to apply judgement of the primary use of the facility, particularly with respect to financial products. 7.3 Structure (Step 8) This eighth step considers the effect of how strongly a facility is structured, its covenants, its conditions, etc., in order to prompt appropriate adjustment(s) to the rating. The lending purposes and/or structure may influence (positively or negatively) the strength and quality of the credit. These may refer to the status of the borrower, the priority of the security, the covenants (or lack thereof) attached to a facility, etc. Take, for example, a facility that has been downgraded due to the term of a loan. If the structure contains very strong covenants which mitigate the effect of the term to maturity of the facility, it may be appropriate to make an adjustment to offset (often partially) the effect of the term to maturity of the facility.

Page 300 Table 7.16 Adjustment to Facility Rating Some instances that might affect ratings are listed in Box 7.4. Other considerations may affect the facility rating. For example, facilities that are readily saleable into the market may merit an upgrade due to their liquidity. 7.4 Collateral (Step 9) This last and ninth step recognizes that the presence of security should heavily affect the severity of loss, given default, in any facility. The quality and depth of security vary widely and will determine the extent of the benefit in reducing any loss. Security should be valued as it would be in a liquidation scenario. In other words, if the business fails, what proceeds would be available? If the total security package includes components from various collateral categories, then one should generally use the worst category containing security on which any significant reliance is placed. The collateral category should reflect only the security held for the facility that is being rated. (Exceptions are where all security is held for all facilities, and where they are being rated as one total.) Documentation risk (the proper completion

Page 301 BOX 7.4 STRUCTURE STRUCTURE ADJUSTMENT Covenants/term: Covenants are in place which effectively mitigate all (or part) of any increased risk due to term, by means of default clauses that provide a full opportunity to make demands, or by means of repayment arrangements that ensure rapid pay-down. ACTION: Upgrade only to offset (possibly partially) any downgrade for term. Poor covenants: Appropriate covenants are not in place, or are very loose, so that review/default may/will not be triggered, even though significant deterioration occurs. ACTION: Downgrade. Subordinated/loans security: The bank's loan is subordinated, putting one's position and/or security significantly behind other creditors. ACTION: Downgrade. Corporate organization: The borrower is highly cash flow dependent on related operating companies that have their own financing. ACTION: Downgrade. of security) is always a concern and should be considered when assessing the level of protection. A few examples of collateral categories are shown in Box 7.5. Collateral can have a major effect on the final facility rating. One should also observe that the value of the collateral is often a function of movements in market rates. Accordingly, the final facility rating is dependent on movement of rates and therefore may be adversely impacted by a significant change in rates. 8 Conclusion The utilization and appropriate processing of a variety of factors (e.g., key financial analysis measures) can provide the credit ana-

Page 302 BOX 7.5 COLLATERAL COLLATERAL CATEGORIES Pledged assets are of very high caliber (generally no reliance on inventory) and provide substantial overcoverage (using conservative valuations, with liquidation appraisals held where warranted). A first charge is held over specific company assets or all company assets (depending on the type of credit facility). Background support may also add strength (personal guarantees do not qualify unless strongly supported). lyst with a tool to arrive at the obligor and facility ratings of a counterparty. The 1999 Basle Conceptual Paper has explicitly recognized that, in the future, an internal risk rating-based system could prove useful to banks in their calculation of the minimum required regulatory capital. 19 Basle has surveyed banks in terms of their methodology, mapping to losses, consistency, oversight, and control as well as internal applications. We would expect that over time more sophisticated banks would all adopt a system based on internal ratings in lieu of a standardized external rating system. Appendix 1 Definitions of Key Ratios

Page 303 Appendix 2 Key Financial Analysis Measures A Liquidity Ability to Meet Short-Term Obligations B Solvency Ability to Meet Key Term Obligations (Ability to Service Debts)

Page 304 C Leverage and Capital Measure D Operating Performance (Profitability of a Business)

Page 305 E Securities Analysis F Ratios for Evaluating the Expenses of a Business G Ratios for Evaluating the Sufficiency of a Firm's Cash Flow H Ratios for Evaluating Collateral Note: If the borrower has more than one loan outstanding, and the loans are owed to the same bank, the balances on all such loans may be combined in the denominator, and the total value of all of the collateral may be combined in the numerator. However, such combinations should never be made if the loans are not explicitly cross-collateralized.

Page 306 Appendix 3A Prototype Industry Assessment: Telecommunications in Canada COMMENTARY ON RISK ASSESSMENT CRITERIA COMPETITIVENESS The Canadian industry's competitive position is favorable with regard to that of its trading partners due to its advanced and technologically up-to-date telecommunications infrastructure. Moreover, changes in the regulatory framework over the past three years have resulted in downward pressure on rates spurred on by the innovative service offerings and pricing plans of the new entrants and the competitive response of the incumbents (especially in the long-distance area). While the erosion of the incumbents' long-distance market share has recently stabilized, the entry of new players in the local market, which was opened to competition on January 1, 1998, will result in renewed losses in market share in local telephony. The move from rate-of-return regulation to a price cap regime will improve the competitiveness of the industry, as the primary means of improving profitability will shift from growing assets to cutting costs and adding new revenue generating services. TRADE ENVIRONMENT Telecommunications services were not included in the NAFTA. Canada agreed to eliminate the monopoly in overseas telephone and fixed domestic satellite services as part of the WTO agreement liberalizing trade in telecommunications. While Canada agreed to remove foreign ownership restrictions in very limited areas (global mobile satellites and submarine cable landings), the 46.7 percent ceiling for telecommunications and broadcast industries was maintained.

Page 307 REGULATORY FRAMEWORK Effective January 1995, the industry's competitive businesses, principally long-distance voice, data, and enhanced services such as ATM and frame relay, have been free of regulation. As of January 1998, the CRTC has opened the local telephone market to competition. By not forcing the incumbent telcos to offer cheap access rates to their local networks, the CRTC eliminates resale as a long-term strategy for the local market in the hope of attracting competitors who are willing to make long-term investments in their own facilities. The incumbent telcos, however, are expected to unbundle their services and provide new entrants with access to local network facilities which they cannot realistically duplicate themselves (e.g., local loops). RESTRUCTURING Driven by major regulatory, technological, and competitive forces, the telecommunications industry has undergone significant restructuring over the past 3 to 4 years. Further restructuring is expected as a result of new entrants in the provision of local telephony. TECHNOLOGICAL CHANGE Technology is what drives this industry and has resulted in the introduction of a host of new enhanced high-margin services. As change is a constant in this industry, substantial capital investments are required and ongoing R&D is imperative if the industry is to keep its competitive edge, and to upgrade its systems to provide the additional services that have been allowed by the CRTC. While the industry as a whole is cash rich and can undertake these expenditures, keeping up with technological change represents a major challenge to smaller telephone systems. FINANCIAL PERFORMANCE Ratios are satisfactory and sustainable. LONG-TERM TRENDS AFFECTING DEMAND Corporate cost-cutting has led to a greater reliance on electronic communications technology, boosting the demand for the industry's services. The increasing use of computers at home will keep demand for telecommunications services high even if some market share in the provision of these services is lost to alternate providers such as the cable companies. VULNERABILITY TO MACROECONOMIC ENVIRONMENTS The industry is mildly affected by the domestic cycle, as consumers may reduce the number of long-distance calls and discontinue some value-added services during a downturn. Source: CIBC Economics Division.

Page 308 Appendix 3B Prototype Industry Assessment: Footwear and Clothing in Canada COMMENTARY ON RISK ASSESSMENT CRITERIA COMPETITIVENESS The apparel industry is dominated by a large number of small firms employing fewer than 50 persons, with very few of these operations benefiting from economies of scale. While some apparel companies are competitive in specific niche markets, such as men's suits and women's lingerie, labor costs in Canada relative to those in low-wage countries leave many apparel operations at a competitive disadvantage. Except in a few specialized areas, Canadian footwear companies are not competitive with the large U.S. operations or the offshore low-cost manufacturers. TRADE ENVIRONMENT All tariffs on Canada-U.S. apparel trade were eliminated on January 1. 1998. All apparel tariffs between Canada and Mexico under NAFTA will be eliminated by January 1, 2003. Under NAFTA, Canadian apparel manufacturers face stricter rules of origin, although in some product cases, duty refunds and tariff preference levels (TPLs) are available. TPLs will be reviewed in 1999. Under the WTO trade rules, Canada has reduced its tariff rates on both footwear and clothing while quantitative restrictions on apparel imports will be eliminated by December 31, 2004. As a result, the Canadian apparel and footwear industries will be facing significantly more import competition in the future. REGULATORY FRAMEWORK Currently, the apparel and footwear industries are not subject to any federal environmental legislation. Some labeling requirements are mandatory, especially if the product is to be exported to a NAFTA country. New simplified (symbols only) U.S. care-labeling rules, when harmonized under the NAFTA, should reduce costs to manufacturers who export within the region.

Page 309 RESTRUCTURING Increased competition from low-cost imports will necessitate further downsizing of the apparel industry. Apparel operations producing standard products that complete directly with low-cost imports will likely close. Further downsizing of leather footwear and skate manufacturing operations is anticipated. TECHNOLOGICAL CHANGE Highly flexible, fast, responsive manufacturing configurations and CAD/CAM design systems allow for more flexibility in terms of product design, layouts for cutting, and short runs. They reduce input waste, as well as labor and inventory costs. Investment in such equipment is difficult to absorb by many of the smaller players in the industry, as is the procurement/hiring of the skilled labor needed to operate this machinery. FINANCIAL PERFORMANCE Overall, ratios are weak and are expected to weaken. Equity levels continue to decline as do profitability ratios. LONG-TERM TRENDS AFFECTING DEMAND With more casual days and flexible working arrangements in business, casual apparel and footwear continues to gain in popularity at the expense of more formal attire (a major portion of Canadian output). This trend is reinforced by the increasing importance of fitness and leisure activities in Canadian lifestyles. VULNERABILITY TO MACROECONOMIC ENVIRONMENTS Both the footwear and apparel industries are highly vulnerable to changes in the Canadian economy. An economic downturn and/or rise in interest rates affects consumer spending. These two industries are also vulnerable to exchange rate movements as many of their inputs are sourced from the United States or offshore. Changes in exchange rates also affect the price of imports, of which most come from low-cost sources. Imports account for 75 percent of Canada's footwear market and 47 percent of Canada's apparel market. Source: CIBC Economics Division.

Page 310 Appendix 4 Prototype Country Analysis Report (Condensed Version): Brazil Country Report: Brazil Date:July 1999 Forecasts Performance Indicators 1996 1997 1998 1999 2000 Positives Nominal GNI US$bn 515 541 496 468 483 GNI per capita US$ 3,262 3,384 3,058 2,847 2,894 Real growth (% ch) 2.8 3.2 0.2 (4.0) (0.5) Investment/GNI ratio (%) 19.4 19.9 17.8 16.0 16.5 Domestic credit growth (%) 25.3 16.0 27.0 11.0 12.0 Unemployment rate (urban centers (%) 5.4 5.7 7.1 11.0 10.8 Inflation rate (% ch) 15.8 6.9 6.1 10.0 20.0 PSBR/GNP ratio (minus sign designates surplus) (%) 6.1 5.9 7.9 11.0 10.0 Total exports US$bn 60.6 67.3 65.1 68.2 71.3 Total exports/gni ratio (%) 11.8 12.4 13.1 14.6 14.8 Merchandise exports US$bn 47.8 53.0 51.1 53.9 57.0 Trade balance US$bn (5.6) (8.4) (6.2) 0.4 (.4) Privatization (utilities, ports, mining; telecoms) Largest market in Latin America IMF-led rescue package from the international community Negatives Economy is in recession Large fast-growing public sector domestic debt (48% of GDP) Still large current account and budget deficits High reliance on foreign investments Needs to Happen Reduce internal debt More structural reforms (i.e. privatization, tax system, social security system) Rapid export growth so that the large external debt can be serviced Current account balance US$bn (23.6) (33.9) (35.2) (29.1) (27.5) Balance of payments US$bn 8.6 (7.6) (8.2) 1.2 1.0 Int'l reserves (gold @ 35 sdr/oz) US$bn 58.6 51.0 42.8 44.0 45.0 Exchange rate (average) Reais/US$ 1.01 1.08 1.16 1.79 2.25 Est. gross financing required US$bn 91.9 112.4 96.2 100.0 91.8 Total external debt US$bn 179.1 182.9 206.4 226.8 256.3 Short-term external debt US$bn 63.8 55.7 38.9 37.1 35.3

Current account/gni ratio (%) 4.6 6.3 7.1 6.2 5.7 External debt/gni ratio (%) 34.8 33.8 41.6 48.5 53.1 External debt/exports ratio (%) 295.7 272.0 316.9 332.6 359.2 Debt service ratio (%) 45.8 66.6 69.7 90.9 82.9 (table continued on next page)

Page 311 (table continued from previous page) Comparative Data 1998 Argentina Chile Mexico Venezuela Real growth % 3.9 3.1 4.6 1.0 Inflation % 0.7 5.0 15.6 36.5 PSBR/GNI % (minus sign denotes surplus) Current account/gni % debt service ratio 1.8 (0.6) 7.5 3.3 (4.8) (6.4) (3.5) (2.2) 49.1 24.4 39.9 29.1 Country Risk Rating Profile Basics Shockability Politics Internal economics External economics High real interest rates, fiscal austerity measures continue to limit the scope for business. Unsatisfactory repayment experience and poor data availability. Pressured by excessive public sector spending and external factors, Brazil was forced to devalue (then float) the real. Foreign savings and IMF are financing budget and current account deficits. Due to the devaluation, the president's popularity is low and his leadership ability has evaporated. The political will to carry out commitments to fiscal adjustment will be tested in the months ahead. Fiscal austerity (to control the public debt and reduce dependence on external savings) and high interest rates amid unfavorable conditions in world markets have led to a deep recession. Due to devaluation of the real, asset prices are now cheaper and this has attracted FDI inflows. After devaluation the current account is adjusting slowly. High financing requirements. Conclusion: The economy and the political system are far from healthy. President Cardoso's leadership ability has evaporated since the January 1999 real devaluation. The political will to carry out important commitments to fiscal adjustment (needed to control the public debt and reduce the economy's dependence on external savings) will be tested in the months ahead. While structural reforms have been approved, they are not being implemented swiftly. Conditions in the financial markets will soon be critical again. Other events or conditions could also trigger problems including rising unrest, as unemployment remains high, political corruption scandals, or domestic debt problems. Source: CIBC Economics Division: Country And International Analysis Group.

Page 312 Notes 1. S&P Corporate Ratings Criteria, 1998, p.3. 2. Moody's Credit Ratings and Research, 1998, p.4. 3. Moody's Investors Service, Rating Methodology: The Evolving Meaning of Moody's Bond Ratings, 1999, p.4. 4. A put bond is a bond stipulation that allows the holder to redeem the bond at face value at a specific, predetermined time so that if interest rates go up the holder can avoid losing money as long as the stipulation is operative; or in other words, it is a bond giving the investor the right to liquidate the bond, or to sell it back to the issuing party. 5. See Nusbaum (1996). 6. The risk of loss is a very general notion since it can be described in several distinct dimensions. For example, one can describe it in relation to the expected loss dimension, the unexpected loss (economic capital) dimension, the 10-bp tail probability of loss dimension, etc. One would need to describe risk of loss in a precise fashion in order to appropriately back-test the degree to which one's RRS was predictive. 7. A typical RRS generally excludes real estate credits, banks, agriculture, public finance, and other identified groups. 8. If, for example, our universe contained 100 rated companies, of which 10 were upgraded during the year and 10 were downgraded, and if the upgraded companies moved on average by 1.5 notches (e.g., five were upgraded by one class and five companies by two risk classes) and if the downgraded firms were all downgraded by one single class, then the rating activity is 25% (= 10 1.5 + 10 1/100), and the drift is 5% (= 10 1.5 10 1/100). 9. The article also presents the five- and 10-year transition matrices. 10. Working capital is defined as the difference between current assets and current liabilities. 11. A clean 100 percent guarantee refers to a straightforward guarantee for 100 percent of the obligation without any conditionality as to the enforceability or collectibility; i.e., the bottom line is that the guarantor is ''on the hook" just as firmly as the original obligor, and has no extra defense under law. 12. For definitions of key accounting ratios, see Appendix 1. 13. As an appropriate control, the average might first be compared to the worst of the three risk levels. The rating should not be more than 1.0 better than the worst rating. In other words, if it exceeds this

Page 313 control, then it must be adjusted downwards. For example, if the three assessment areas were respectively rated 2, 2, and 5, then the average is 3, but the rating should be adjusted to 4 (being 1.0 better than the 5 risk level). If the worst of the three risk levels is not an integer (say 4.5), then reducing by 1 would leave a 3.5. One typically uses judgement to set the rating at either a 3 or 4. 14. Business risk is defined as the risk associated with the level and stability of operating cash flows over time. Further detail can be found in Chapter 17. 15. A company can create working capital by borrowing on a long-term basis and employing the proceeds of the loan for current assets. Working capital will increase by the amount of additional long-term debt less any addition to current liabilities. 16. Working capital is sometimes created by the sale of fixed assets and it increases by the exact amount of the reduction of fixed assets. As companies grow, however, it is more likely that the fixed assets in the formula will represent a competing use of the various working capital sources 17. A keepwell agreement is an agreement in which one party agrees to maintain a certain status or condition at another company; e.g., a parent company may agree to maintain the net worth of a subsidiary company at a certain level. This is a legally enforceable contract; however, only the party that the keepwell is in favor of may sue under such a contract. 18. A comfort letter is a letter generally requested by securities underwriters to give comfort on the financial information included in an SEC registration statement. It generally expresses the support of the parent company for the activities and commitments of subsidiaries. However, it does not constitute a guarantee. 19. See also Chapter 2.

Page 315 Chapter 8 The Credit Migration Approach to Measuring Credit Risk 1 Introduction With the Bank for International Settlements (BIS) 1998 reform in place, internal models for both general and specific market risk have been implemented at the major G-10 banks and are used every day to report regulatory capital for the trading book. The next step, as we discussed in Chapter 2, is to extend the VaR framework so that it can be used to allocate capital for credit risk in the banking book. The current BIS requirements for the "specific risk" component of market risk are subject to broad interpretation (Chapter 4). 1 To qualify as an internal model for specific risk, the regulator should be convinced that "concentration risk," "spread risk," "downgrade risk,'' and "default risk" are appropriately captured. The exact meaning of "appropriately" is left to the appreciation of both the bank and the regulator. The capital charge for specific risk is then the product of a multiplier (the minimum value of which is currently set to 4) times the sum of the VaR at the 99 percent confidence level for spread risk, downgrade risk, and default risk over a 10-day horizon. There are several problems with this piecemeal approach to credit risk. First, spread risk is related to both market risk and credit risk. Spreads fluctuate either because equilibrium conditions in

Page 316 capital markets change, which in turn affects credit spreads for all credit ratings, or because the credit quality of the obligor has improved or deteriorated, or because both conditions have occurred simultaneously. In addition, spreads depend on the liquidity of the market for corporate bonds. Downgrade risk, on the other hand, is a pure credit spread risk. When the credit quality of an obligor deteriorates, then the spread relative to the Treasury curve widens, and vice versa when the credit quality improves. Simply adding spread risk to downgrade risk may lead to double counting. In addition, the current regime incorporates the market risk component of spread risk into the calculation for credit risk. This means that the regulatory capital multiplier for this risk is 4 instead of 3. Second, the problem of disentangling the market risk and credit risk driven components in spread changes is further obscured by the fact that often market participants anticipate forthcoming credit events before they actually happen. Therefore, spreads already reflect the new credit status by the time the rating agencies effectively downgrade an obligor, or put the obligor on "credit watch." Third, default is simply a special case of downgrade: the credit quality has deteriorated to the point at which the obligor cannot service its debt obligations. An adequate credit VaR model should therefore address migration risk (i.e., credit spread risk) and default risk within a consistent and integrated framework. Finally, changes in market and economic conditions, as reflected by changes in interest rates, stock market indexes, exchange rates, unemployment rates, etc., may affect the overall profitability of firms. As a result, the exposures of the various counterparties to each obligor, as well as the probabilities of default and of migrating from one credit rating to another, are linked to market risks. Thus, in an ideal world, the risk framework would fully integrate market risk and credit risk. Over the last few years, a number of new approaches to credit modeling have been made public. CreditMetrics from JP Morgan, first published in 1997, is reviewed in the next section of this chapter. The CreditMetrics approach is based on the analysis of credit migration, i.e., the probability of moving from one credit quality

Page 317 to another, including default, within a given time horizon (often arbitrarily taken to be one year). CreditMetrics models the full forward distribution of the values of any bond or loan portfolio, say one year forward, where the changes in values are related to credit migration only; interest rates are assumed to evolve in a deterministic fashion. The credit VaR of a portfolio is then derived in a similar fashion as for market risk. It is simply the distance from the mean of the percentile of the forward distribution, at the desired confidence level. (This definition applies to all credit risk models, and is independent of the underlying theoretical framework.) Tom Wilson (1997a, 1997b) proposes an improvement to the credit migration approach by allowing default probabilities to vary with the credit cycle. In this approach, default probabilities are a function of macrovariables such as unemployment, the level of interest rates, the growth rate in the economy, government expenses, and foreign exchange rates. These macrovariables are the factors that, to a large extent, drive credit cycles. This methodology is reviewed in Section 3 of this chapter. Over the last few years, KMV Corporation, a firm that specializes in credit risk analysis, has developed a credit risk methodology and extensive database to assess default probabilities and the loss distribution related to both default and migration risks. KMV's methodology differs somewhat from CreditMetrics, in that it relies upon the "expected default frequency," or EDF, for each issuer, rather than upon the average historical transition frequencies produced by the rating agencies for each credit class. The KMV approach is based on the asset value model originally proposed by Merton (1974), the main difference being the simplifying assumptions required to facilitate the model's implementation. It remains to be seen whether these compromises prevent the model from capturing the real complexity of credit. KMV's methodology, together with the contingent claim approach to measuring credit risk, is reviewed in Chapter 9. 2 At the end of 1997, Credit Suisse Financial Products (CSFP) released an approach that is based on actuarial science. CreditRisk+, which focuses on default alone rather than credit migration, is examined briefly in Chapter 10. CreditRisk+ assumes that the dynamics of default for individual bonds, or loans, follow a Poisson process.

Page 318 Finally, the "reduced form" approach currently the foundation of credit derivative pricing models is reviewed in Chapter 11. These models allow one to derive the term structure of default probabilities from credit spreads, while assuming an exogenous and somewhat arbitrary recovery rate. In Chapter 12 we compare CreditMetrics, KMV, CreditRisk+, and the BIS 1988 standardized approach for a benchmark portfolio composed of more than 1800 bonds. The bonds we chose were diversified across 13 currencies covering North America, Europe, and Asia; across various maturities; and across the whole spectrum of credit qualities. It appears that credit VaR numbers generated by these three models fall within a narrow range, with a ratio of 1.5 between the highest and the lowest values, provided consistent inputs across models are used. Thus although default is modeled differently, and in some instances (e.g., CreditRisk+) downgrade risk is ignored, the models produce relatively close results for the same portfolio. This is not that surprising, since the main risk factor is default. However, it is interesting to note that when the portfolio contains a large proportion of bonds from OECD banks, the standardized approach produces a lower capital charge than the credit models. This is because the standardized approach allots very low risk weights to those institutions that do not always reflect their actual credit risk. But if we assume that all bonds are investment-grade corporate bonds, the credit models generate a capital charge that is substantially lower than that based on the 1988 BIS Accord. It seems that any of these three models can be considered as a reasonable internal model to assess regulatory capital related to credit risk for straightforward bonds and loans without option features. However, all the models assume deterministic interest rates. This means that the models are inappropriate for measuring the credit risk of swaps, credit derivatives, and other derivative-like products such as loan commitments. 3 For these instruments we need an integrated framework that permits the derivation of the credit exposure and the loss distribution in a consistent manner. Currently, none of the proposed models offers such an integrated approach. In order to measure the credit risk of derivative securities, the next generation of credit models should allow for stochastic interest rates, and possibly for

Page 319 Figure 8.1 Corporate Defaults Worldwide (Number of Firms and the Amount Defaulted) Source: Standard & Poor's. default and migration probabilities that depend on the state of the economy. In Figure 8.1 we present the record of defaults from 1985 to 1997. In 1990 and 1991, when the world economies were in recession, the frequency of defaults increased substantially. During the post-1991 period, which have been characterized by a "growth economy," the default rate has declined dramatically. Only recently the default rate has started to rise. 2 Creditmetrics Framework 4 CreditMetrics is a methodology that is based on the estimation of the forward distribution of changes in the value of a portfolio of loan- and bond-type products at a given time horizon, usually one year. 5 The changes in value are related to the migration, upwards and downwards, of the credit quality of the obligor, as well as to default.

Page 320 In comparison to market VaR, credit VaR poses three challenges. First, the portfolio distribution is far from being a normal distribution. Second, measuring the portfolio effect due to diversification is much more complex than for market risk. Third, the information on loans is not as complete as it is for traded instruments such as bonds. While it may be reasonable to assume that changes in portfolio values are normally distributed when due to market risk, credit returns are by their nature highly skewed and fat-tailed (Figure 8.2). An improvement in credit quality brings limited "upside" to an investor, while downgrades or defaults bring with them substantial "downsides." Unlike market VaR, the percentile levels of the distribution cannot be estimated from the mean and variance only. The calculation of VaR for credit risk thus demands a simulation of the full distribution of the changes in the value of the portfolio. To measure the effect of portfolio diversification, we need to estimate the correlations in credit quality changes for all pairs of Figure 8.2 Comparison of the Probability Distributions of Credit Returns and Market Returns Source: CIBC.

Page 321 obligors. However, these correlations are not directly observable. CreditMetrics bases its evaluation on the joint probability of equity returns. This entails making some strong simplifying assumptions about the capital structure of the obligor, and about the process that is generating equity returns. We will elaborate on this key feature of the model later on. Finally, as we mentioned above, CreditMetrics, like KMV and CreditRisk+, makes no provision for market risk: forward values and exposures are derived from deterministic forward curves. The only uncertainty in CreditMetrics relates to credit migration, i.e., the process of moving up or down the credit spectrum. The CreditMetrics risk measurement framework is summarized in Figure 8.3, which shows the two main building blocks: 1. "Value at Risk Due to Credit" for a single financial instrument. 2. Value at Risk at the portfolio level, which accounts for portfolio diversification effects ("Portfolio Value at Risk Due to Credit"). There are also two supporting functions. "Correlations" derives the equity return correlations used to generate the joint migration probabilities; "exposures" produces the future exposures of derivative securities, such as swaps. 3 Credit VaR for a Bond (Building Block 1) The first step is to specify a rating system, with rating categories, together with the probabilities of migrating from one credit quality to another over the credit risk horizon. This transition matrix is the key component of the credit VaR model proposed by JP Morgan. The matrix may take the form of the rating system of Moody's, or Standard & Poor's, or it might be based on the proprietary rating system internal to the bank. A strong assumption made by CreditMetrics is that all issuers within the same rating class are homogeneous credit risks. They have the same transition probabilities and the same default probability. (KMV departs from CreditMetrics in the sense that in KMV's framework each issuer is specific, and is characterized by its own asset

Page 322 returns distribution, its own capital structure, and its own default probability.) Second, the risk horizon should be specified. This is usually taken to be one year. When one is concerned about the risk profile over a longer period of time, as for long-dated illiquid instruments, multiple horizons can be chosen, such as one to 10 years. The third step consists of specifying the forward discount curve at the risk horizon(s) for each credit category. In the case of default, the value of the instrument should be estimated in terms of the "recovery rate," which is given as a percentage of face value or "par." In the final step, this information is translated into the forward distribution of the changes in the portfolio value consecutive to credit migration. Figure 8.3, taken from the technical document of CreditMetrics, illustrates the steps of the credit VaR model. In their example we Figure 8.3 CreditMetrics Framework: The Four Building Blocks Source: CreditMetrics, JP Morgan.

Page 323 calculate credit VaR for a senior unsecured BBB rated bond maturing in exactly five years, and paying an annual coupon of 6 percent. Step 1 Specify the Transition Matrix The rating categories, as well as the transition matrix, are chosen from an external or internal rating system (Table 8.1). In the case of Standard & Poor's, there are seven rating categories. The highest credit is AAA; the lowest, CCC. Default is defined as a situation in which the obligor cannot make a payment related to a bond or a loan obligation, whether the payment is a coupon payment or the redemption of the principal. "Pari passu" clauses are such that when an obligor defaults on one payment related to a bond or a loan, the obligor is technically declared in default on all debt obligations. The bond issuer in our example is currently a BBB rating. The BBB row in Table 8.1 shows the probability, as estimated by Standard & Poor's, that this BBB issuer will migrate over a period of one year to any one of the eight possible states, including default. Obviously, the most probable situation is that the obligor will remain in the same rating category, i.e., BBB; this has a probability of 86.93 percent. The probability of the issuer defaulting within one TABLE 8.1 Transition Matrix: Probabilities of Credit Rating Migrating From One Rating Quality to Another, Within One Year Initial Rating Rating at Year-End (%) AAA AA A BBB BB B CCC Default AAA 90.81 8.33 0.68 0.06 0.12 0 0 0 AA 0.70 90.65 7.79 0.64 0.06 0.14 0.02 0 A 0.09 2.27 91.05 5.52 0.74 0.26 0.01 0.06 BBB 0.02 0.33 5.95 86.93 5.30 1.17 1.12 0.18 BB 0.03 0.14 0.67 7.73 80.53 8.84 1.00 1.06 B 0 0.11 0.24 0.43 6.48 83.46 4.07 5.20 CCC 0.22 0 0.22 1.30 2.38 11.24 64.86 19.79 Source: Standard & Poor's CreditWeek (April 15, 1996).

year is only 0.18 percent, while the probability of it being upgraded to AAA is also very small, i.e., 0.02 percent. Such a transition matrix is produced by the rating agencies for all initial ratings, based on the history of credit events that have occurred to the firms rated by those agencies. (Default is taken to be an "absorbing state"; i.e., when an issuer is in default, it stays in default.) Page 324 Moody's publishes similar information. The probabilities published by the agencies are based on more than 20 years of data across all industries. Obviously, these data should be interpreted with care since they represent average statistics across a heterogeneous sample of firms, and over several business cycles. For this reason many banks prefer to rely on their own statistics, which relate more closely to the composition of their loan and bond portfolios. Moody's and Standard & Poor's also produce long-term average cumulative default rates, as shown in Table 8.2 in a tabular form and in Figure 8.4 in graphical form. For example, a BBB issuer has a probability of 0.18 percent of defaulting within one year, 0.44 percent of defaulting in two years, 4.34 percent of defaulting in 10 years, and so on. Tables 8.1 and 8.2 should, in fact, be consistent with one another. From Table 8.2 we can derive the transition matrix that best replicates the average cumulative default rates. Indeed, assuming TABLE 8.2 Average Cumulative Default Rates (Percent) Term 1 2 3 4 5 7 10 15 AAA 0.00 0.00 0.07 0.15 0.24 0.66 1.40 1.40 AA 0.00 0.02 0.12 0.25 0.43 0.89 1.29 1.48 A 0.06 0.16 0.27 0.44 0.67 1.12 2.17 3.00 BBB 0.18 0.44 0.72 1.27 1.78 2.99 4.34 4.70 BB 1.06 3.48 6.12 8.68 10.97 14.46 17.73 19.91 B 5.20 11.00 15.95 19.40 21.88 25.14 29.02 30.65 CCC 19.79 26.92 31.63 35.97 40.15 42.64 45.10 45.10 Source: Standard & Poor's CreditWeek (April 15, 1996).

Page 325 Figure 8.4 Average Cumulative Default Rates (%) by Rating Source: Standard & Poor's CreditWeek (April 15, 1996). that the process for default is Markovian and stationary, then multiplying the one-year transition matrix n times generates the n-year matrix. The n-year default probabilities are simply the values in the last default column of the transition matrix, and should match the column in year n of Table 8.2. 6 The realized transition and default probabilities vary quite substantially over the years, depending upon whether the economy is in recession or is expanding (Figure 8.1). When implementing a model that relies on transition probabilities, one may have to adjust the average historical values, as shown in Table 8.1, to be consistent with one's assessment of the current economic environment. A study provided by Moody's (Carty and Lieberman, 1996) provides historical default statistics, both the mean and standard deviation, by rating category for the population of obligors that they rated during the period 1970 to 1995 (Table 8.3).

Page 326 TABLE 8.3 One-Year Default Rates by Rating, 1970 1995 One-Year Default Rate Credit Rating Average (%) Standard Deviation (%) Aaa 0.00 0.0 Aa 0.03 0.1 A 0.01 0.0 Baa 0.13 0.3 Ba 1.42 1.3 B 7.62 5.1 Source: Carty and Lieberman (1996). Step 2 Specify the Credit Risk Horizon The risk horizon is usually set at one year, and is consistent with the transition matrix shown in Table 8.1. But this horizon is arbitrary, and is mostly dictated by the availability of the accounting data and the financial reports processed by the rating agencies. In KMV's framework, which relies on market data as well as accounting data, any horizon can be chosen from a few days to several years. Indeed, market data can be updated daily; it is assumed that the other characteristics of the borrowers remain constant (until new information for these, too, becomes available). Step 3 Specify the Forward Pricing Model The valuation of a bond is derived from the zero curve corresponding to the rating of the issuer. Since there are seven possible credit qualities, seven ''spread" curves are required to price the bond in all possible states. All obligors within the same rating class are then marketto-market using the same curve. The spot zero curve is used to determine the current spot value of the bond. The forward price of the bond one year from the present is derived from the forward zero curve, one year ahead, which is then applied to the residual

Page 327 TABLE 8.4 One-Year Forward Zero Curves for Each Credit Rating (%) Category Year 1 Year 2 Year 3 Year 4 AAA 3.60 4.17 4.73 5.12 AA 3.65 4.22 4.78 5.17 A 3.72 4.32 4.93 5.32 BBB 4.10 4.67 5.25 5.63 BB 5.55 6.02 6.78 7.27 B 6.05 7.02 8.03 8.52 CCC 15.05 15.02 14.03 13.52 Source: CreditMetrics, JP Morgan. cash flows from year 1 to the maturity of the bond. Table 8.4 gives the one-year forward zero curves for each credit rating. Empirical evidence shows that for high-grade investment bonds, the spreads tend to increase with time to maturity, while for low-grade bonds (e.g., CCCs) the spread tends to be wider at the short end of the curve than at the long end, as shown in Figure 8.5. Figure 8.5 Spread Curves for Different Credit Qualities

Page 328 If the obligor remains rated BBB, the one-year forward price, V BBB, of the five-year, 6 percent coupon bond (Box 8.1) is: Box 8.1 If we replicate the same calculations for each rating category, we obtain the values shown in Table 8.5. 7 We do not assume that everything is lost if the issuer defaults at the end of the year. Depending on the seniority of the instrument, a recovery rate of par value is recuperated by the investor. TABLE 8.5 One-Year Forward Values for a BBB Bond Year-End Rating Value ($) AAA 109.37 AA 109.19 A 108.66 BBB 107.55 BB 102.02 B 98.10 CCC 83.64 Default 51.13 Source: CreditMetrics, JP Morgan.

Page 329 TABLE 8.6 Recovery Rates by Seniority Class (Percent of Face Value, i.e., "Par") Seniority Class Mean (%) Standard Deviation (%) Senior Secured 53.80 26.86 Senior Unsecured 51.13 25.45 Senior Subordinated 38.52 23.81 Subordinated 32.74 20.18 Junior Subordinated 17.09 10.90 Source: Carty and Lieberman (1996). These recovery rates are estimated from historical data by the rating agencies. Table 8.6 shows the expected recovery rates for bonds by different seniority classes as estimated by Moody's. 8 In our example the recovery rate for senior unsecured debt is estimated to be 51.13 percent, although the estimation error is quite large (s.d. = 25.45 percent) and the actual value lies in a fairly large confidence interval. When the loss distribution is derived from a Monte Carlo simulation, it is generally assumed that the recovery rates are distributed according to a beta distribution with the same mean and standard deviation as shown in Table 8.6. Step 4 Derive the Forward Distribution of the Changes in Portfolio Value The distribution of the changes in the bond value, at the one-year horizon, due to an eventual change in credit quality is shown Table 8.7 and Figure 8.6. This distribution exhibits a long "downside tail." The first percentile of the distribution of V, which corresponds to credit VaR at the 99 percent confidence level, is 23.91. It is a much lower value than if we computed the first percentile assuming a normal distribution for V. In that case credit VaR at the 99 percent confidence level would be only 7.43. 9

Page 330 TABLE 8.7 Distribution of the Bond Values, and Changes in Value of a BBB Bond, in One Year Year-End Rating Probability of State: p (%) Forward Price: V ($) Change in Value: V ($) AAA 0.02 109.37 1.82 AA 0.33 109.19 1.64 A 5.95 108.66 1.11 BBB 86.93 107.55 0 BB 5.30 102.02 5.53 B 1.17 98.10 9.45 CCC 0.12 83.64 23.91 Default 0.18 51.13 56.42 Source: CreditMetrics, JP Morgan. 4 Credit VaR for a Loan or Bond Portfolio (Building Block 2) First, consider a portfolio composed of two bonds with an initial rating of BB and A, respectively. Given the transition matrix shown in Table 8.1, and assuming no correlation between changes in credit quality, we can then derive easily the joint migration probabilities shown in Table 8.8. Each entry is simply the product of the transition probabilities for each obligor. For example, the joint probability that obligor 1 and obligor 2 will stay in the same rating class is the product of the probability of bond A remaining at its current rating at the end of the year, 91.05 percent, and the probability of bond BB remaining as BB, 80.53 percent: Unfortunately, when we need to assess the diversification effect on a large loan or bond portfolio, this table is not very useful in practice. In reality, the correlations between the changes in credit quality are not zero. And it will be shown in Section 5 that the overall credit VaR is quite sensitive to these correlations. Their accurate estimation is therefore one of the key determinants of portfolio optimization.

Figure 8.6 Histogram of the One-Year Forward Prices and Changes in Value of a BBB Bond Page 331

Page 332 TABLE 8.8 Joint Migration Probabilities (Percent) with Zero Correlation for Two Issuers Rated BB and A Obligor 2 Obligor 1 (BB) AAA 0.09 AA 2.27 A 91.05 BBB 5.52 BB 0.74 B 0.26 CCC 0.01 Default 0.06 AAA 0.03 0.00 0.00 0.03 0.00 0.00 0.00 0.00 0.00 AA 0.14 0.00 0.00 0.13 0.01 0.00 0.00 0.00 0.00 A 0.67 0.00 0.02 0.61 0.40 0.00 0.00 0.00 0.00 BBB 7.73 0.01 0.18 7.04 0.43 0.06 0.02 0.00 0.00 BB 80.53 0.07 1.83 73.32 4.45 0.60 0.20 0.01 0.05 B 8.84 0.01 0.20 8.05 0.49 0.07 0.02 0.00 0.00 CCC 1.00 0.00 0.02 0.91 0.06 0.01 0.00 0.00 0.00 Default 1.06 0.00 0.02 0.97 0.06 0.01 0.00 0.00 0.00 Default correlations might be expected to be higher for firms within the same industry or in the same region than for firms in unrelated sectors. In addition, correlations vary with the relative state of the economy in the business cycle. If there is a slowdown in the economy, or a recession, most of the assets of the obligors will decline in value and quality, and the likelihood of multiple defaults increases substantially. The opposite happens when the economy is performing well: default correlations go down. Thus, we cannot expect default and migration probabilities to stay stationary over time. There is clearly a need for a structural model that relates changes in default probabilities to fundamental variables. Both CreditMetrics and KMV derive the default and migration probabilities from a correlation model of the firm's assets, as detailed in the next chapter. For the sake of simplicity, CreditMetrics makes use of the stock price of a firm as a proxy for its asset value as the true asset value is not directly observable. (This is another simplifying assumption in CreditMetrics that may affect the accuracy of the approach.) CreditMetrics estimates the correlations between the equity returns of various obligors, and then it infers the correlations between changes in credit quality directly from the joint distribution of these equity returns.

Page 333 TABLE 8.9 Balance Sheet of Merton's Firm Assets Liabilities/Equity Risky Assets: V t Debt: B t (F) Equity: S t Total: V t V t The theoretical framework underlying all this is the option pricing approach to the valuation of corporate securities first developed by Merton (1974). The basic model is presented in Appendix 1 of this chapter, and it is described in detail in Chapter 9 as it forms the basis for the KMV approach. In Merton's model, the firm is assumed to have a very simple capital structure; it is financed by equity, S t, and a single zero-coupon debt instrument maturing at time T, with face value F, and current market value B t. The firm's balance sheet is represented in Table 8.9, where V t is the value of all the assets, and V t = B t (F) + S t. In this framework, default occurs at the maturity of the debt obligation only when the value of assets is less than the promised payment, F, to the bondholders. Figure 8.7 shows the distribution of the assets' value at time T, the maturity of the zero-coupon debt, and the probability of default (i.e., the shaded area on the left-hand side of the default point, F). Merton's model is extended by CreditMetrics to include changes in credit quality as illustrated in Figure 8.8. This generalization consists of slicing the distribution of asset returns into bands in such a way that, if we draw randomly from this distribution, we reproduce exactly the migration frequencies as shown in the transition matrices that we discussed earlier. Figure 8.8 shows the distribution of the normalized assets' rates of return, one year ahead. The distribution is normal with a mean of zero and a variance of 1. The credit rating "thresholds" correspond to the transition probabilities in Table 8.10 for a BB-rated obligor. The right tail of the distribution, down to Z AAA, corresponds to the probability that the obligor will be upgraded

Page 334 Figure 8.7 Distribution of the Firm's Asset Value at Maturity of the Debt Obligation Figure 8.8 Generalization of the Merton Model to Include Rating Changes

Page 335 TABLE 8.10 Transition Probabilities and Credit Quality Thresholds for BB and A Rated Obligors A Rated Obligor BB Rated Obligor Rating in One Year Probabilities (%) Thresholds: Z (σ) Probabilities (%) Thresholds: Z (σ) AAA 0.09 3.12 0.03 3.43 AA 2.27 1.98 0.14 2.93 A 91.05 1.51 0.67 2.39 BBB 5.52 2.30 7.73 1.37 BB 0.74 2.72 80.53 1.23 B 0.26 3.19 8.84 2.04 CCC 0.01 3.24 1.00 2.30 Default 0.06 1.06 from BB to AAA, i.e., 0.03 percent. Then, the area between Z AA and Z AAA corresponds to the probability of being upgraded from BB to AA, etc. The left tail of the distribution, on the left-hand side of Z CCC, corresponds to the probability of default, i.e., 1.06 percent. Table 8.10 shows the transition probabilities for two obligors rated BB and A, respectively, and the corresponding credit quality thresholds. The thresholds are given in terms of normalized standard deviations. For example, for a BB rated obligor the default threshold is 2.30 standard deviations from the mean rate of return. This generalization of Merton's model is quite easy to implement. It assumes that the normalized log-returns over any period of time are normally distributed with a mean of 0 and a variance of 1, and that the distribution is the same for all obligors within the same rating category. If Prob(Def) denotes the probability of the BB rated obligor defaulting, then the critical asset value V Def is such that: which can be translated into a normalized threshold Z CCC, such that the area in the left tail below Z CCC is Prob(Def). 10 Z CCC is sim-

Page 336 ply the threshold point in the standard normal distribution, N(0,1), corresponding to a cumulative probability of Prob(Def). Then, based on the option pricing model, the critical asset value V Def which triggers default is such that Z CCC = d 2. This critical asset value is also called the ''default point." 11 Note that only the threshold levels are necessary to derive the joint migration probabilities, and these can be calculated without it being necessary to observe the asset value, and to estimate its mean and variance. To derive the critical asset value V Def we only need to estimate the expected asset return µ and asset volatilityσ. Accordingly, Z B is the threshold point corresponding to a cumulative probability of being either in default or in rating CCC, i.e., Prob(Def) + Prob(CCC), etc. We mentioned above that, as asset returns are not directly observable, CreditMetrics makes use of equity returns as their proxy. Yet using equity returns in this way is equivalent to assuming that all the firm's activities are financed by means of equity. This is a major drawback of the approach, especially when it is being applied to highly leveraged companies. For those companies, equity returns are substantially more volatile, and possibly less stationary, than the volatility of the firm's assets. Now, assume that the correlation between the assets' rates of return is known, and is denoted byρ, which is assumed to be equal to 0.20 in our example. The normalized log-returns on both assets follow a joint normal distribution, as described in Appendix 1. We can then compute the probability for both obligors of being in any particular combination of ratings. For example, we can compute the probability that they will remain in the same rating classes, i.e., BB and A, respectively: where r BB and r A are the instantaneous rates of return on the assets of obligors BB and A, respectively. If we implement the same procedure for the other 63 combinations, we obtain Table 8.11. We can now compare Table 8.11 with Table 8.8, which was derived under the assumption that there was zero correlation between the companies. Notice that the joint probabilities are different. Figure 8.9 illustrates the effect of asset return correlation on the joint default probability for the rated BB and A obligors. If the prob-

Page 337 TABLE 8.11 Joint Rating Probabilities (Percent) for BB and A Rated Obligors When Correlation Between Asset Returns is 20 Percent Rating of First Company (BB) Rating of Second Company (A) AAA AA A BBB BB B CCC Def Total AAA 0.00 0.00 0.03 0.00 0.00 0.00 0.00 0.00 0.03 AA 0.00 0.01 0.13 0.00 0.00 0.00 0.00 0.00 0.14 A 0.00 0.04 0.61 0.01 0.00 0.00 0.00 0.00 0.67 BBB 0.02 0.35 7.10 0.20 0.02 0.01 0.00 0.00 7.69 BB 0.07 1.79 73.65 4.24 0.56 0.18 0.01 0.04 80.53 B 0.00 0.08 7.80 0.79 0.13 0.05 0.00 0.01 8.87 CCC 0.00 0.01 0.85 0.11 0.02 0.01 0.00 0.00 1.00 Def 0.00 0.01 0.90 0.13 0.02 0.01 0.00 0.00 1.07 Total 0.09 2.29 91.06 5.48 0.75 0.26 0.01 0.06 100 Source: CreditMetrics, JP Morgan. abilities of default for obligors rated A and BB are Prob(DefA) = 0.06 percent and Prob(DefBB) = 1.06 percent, respectively, and the correlation coefficient between the rates of return on the two assets is ρ = 20 percent, it can be shown that the joint probability of default is: The correlation coefficient between the two default events is: 12 Asset returns correlations are approximatively 10 times larger than default correlations for asset correlations in the range of 20 to 60 percent. This shows that the joint probability of default is in fact quite sensitive to pair-wise asset return correlations, and it illustrates how important it is to estimate these data correctly if one is to assess the diversification effect within a portfolio. In Section 5 we show that, for the benchmark portfolio we selected for the comparison of credit models, the impact of correlations on credit VaR is quite large. It is larger for portfolios with relatively low-grade credit quality than it is for high-grade portfolios. Indeed, as the

Page 338 Figure 8.9 Probability of Joint Defaults as a Function of Asset Return Correlation Source: CreditMetrics, JP Morgan. credit quality of the portfolio deteriorates and the expected number of defaults increases, this number is magnified by an increase in default correlations. (The statistical procedure employed to estimate asset return correlations is discussed in the next chapter. 13 ) 5 Analysis of Credit Diversification (Building Block 2, Continuation) The analytic approach that we sketched out above for a portfolio with bonds issued by two obligors is not practicable for large portfolios. Instead, CreditMetrics implements a Monte Carlo simulation to generate the full distribution of the portfolio values at the credit horizon of one year. The following steps are necessary: Derivation of the asset return thresholds for each rating category. Estimation of the correlation between each pair of obligors' asset returns.

Page 339 Generation of asset return scenarios according to their joint normal distribution. A standard technique that is often used to generate correlated normal variables is the Cholesky decomposition. 14 Each scenario is characterized by n standardized asset returns, one for each of the n obligors in the portfolio. For each scenario, and for each obligor, the standardized asset return is mapped into the corresponding rating, according to the threshold levels derived in step 1. Given the spread curves, which apply for each rating, the portfolio is revalued. Repeat the procedure a large number of times, say 100,000 times, and plot the distribution of the portfolio values to obtain a graph (which will look like Figure 8.2). Then, we can derive the percentiles of the distribution of the future values of the portfolio. 6 Credit VaR and Calculation of the Capital Charge Economic capital is the financial cushion that a bank uses to absorb unexpected losses, e.g., those related to credit events such as credit migration and/or default. Figure 8.10 illustrates how the capital charge related to credit risk can be derived: P(c) FV V 0 = value of the portfolio in the worst case scenario at the (I c)% confidence level = forward value of the portfolio = V 0 (1 + PR) = current mark-to-market value of the portfolio PR = promised return on the portfolio 15 EV ER EL = expected value of the portfolio = V 0 (1 + ER) = expected return on the portfolio = expected loss = FV EV The expected loss does not contribute to the capital allocation. Instead, funds equivalent to the expected loss are held in the bank's reserves and the expected loss figure is imputed as a cost into any risk-adjusted return on capital (RAROC) calculation that the bank

Page 340 Figure 8.10 Credit VaR and Calculation of Economic Capital wishes to perform. 16 The capital charge is instead a function of the unexpected losses: The bank should hold reserves against these unexpected losses at a given confidence level, say, 1 percent, so that there is only a probability of 1 percent that the bank will incur losses above the capital level over the period corresponding to the credit risk horizon, say, one year. 7 Creditmetrics as a Loan/Bond Portfolio Management Tool: Marginal Risk Measures (Building Block 2, Continuation) In addition to the overall credit VaR analysis for the portfolio, CreditMetrics offers the interesting feature of isolating the individual marginal risk contributions to the portfolio. For example, for each asset, CreditMetrics calculates the marginal standard deviation, i.e., the impact of each individual asset on the overall portfolio standard deviation. By comparing the marginal standard deviation to the stand-alone

Page 341 Figure 8.11 Risk versus Size of Exposures within a Typical Credit Portfolio Source: CreditMetrics, JP Morgan. standard deviation for each loan, one can assess the extent of the benefit derived from portfolio diversification when adding the instrument in the portfolio. Figure 8.11 shows the marginal standard deviation for each asset, expressed as a percentage of the overall standard deviation, plotted against the marked-to-market value of the instrument. This is an important proactive risk management tool, as it allows one to identify trading opportunities in the loan/bond portfolio where concentration, and as a consequence overall risk, can be reduced without affecting expected profits. Obviously, for this framework to become fully operational it needs to be complemented by a RAROC model that provides information on the adjusted return on capital for each deal. 17 The same framework can also be used to set up credit risk limits and monitor credit risk exposures in terms of the joint combination of market value and marginal standard deviation, as shown in Figure 8.12.

Page 342 Figure 8.12 Possible Risk Limits for an Example Portfolio Source: CreditMetrics, JP Morgan. 8 Estimation of Asset Correlations (Building Block 3) As we discussed above, default correlations are derived from asset returns correlations, which in turn are proxied by equity returns correlations. For a large portfolio of bonds and loans, with thousands of obligors, this still requires the computation of a huge correlation matrix for each pair of obligors. To reduce the dimensionality of this estimation problem, CreditMetrics uses multifactor analysis. This approach maps each obligor to the countries and industries that are most likely to determine its performance. Equity returns are correlated to the extent that firms are exposed to the same industries and countries. To implement CreditMetrics, the user specifies the industry and country weights for each obligor, as well as the "firm specific risk," which is idiosyncratic to each obligor and is not correlated to any other obligor or to any index. 18

Page 343 9 Exposures (Building Block 4) The term "exposures" in the CreditMetrics methodology is somewhat misleading since, in fact, the approach assumes that market risk factors are constant. So the "exposures" building block is, in fact, simply the term CreditMetrics gives to the use of the forward pricing model applied to each credit rating. For bond-type products such as bonds, loans, receivables, commitments to lend, and letters of credit, "exposure'' simply relates to the future cash flows at risk beyond the one-year horizon. The forward pricing is derived from the present value model using the forward yield curve for the corresponding credit quality. The example presented in Section 2 illustrated how the exposure distribution is calculated for a bond. For derivatives, such as swaps and forwards, the exposure is conditional on future interest rates. For these instruments, there is no simple way to derive the future cash flows at risk without making assumptions about the dynamics of interest rates. The position is complicated because the risk exposure for a derivative is dynamic. For example, the credit risk exposure of a swap can be positive (if the swap is in-the-money for the bank) or negative (if it is out-of-the money). In the latter case, the swap is a liability to the bank and the swap counterparty is the only party to suffer credit risk. Figure 8.13 shows the average and maximum exposure profiles of an interest rate swap as a function of time, assuming no change in the credit ratings of the counterparty or the bank. The bank is at risk only when the exposure is positive. CreditMetrics assumes as given the average exposure of a swap, which is supposed to have been derived from an external model. In CreditMetrics, as the interest rates are deterministic, the calculation of the forward price distribution relies on an ad hoc procedure: expected loss in years one to maturity for the given rating R where

Page 344 Figure 8.13 Risk Exposure of an Interest Rate Swap The forward risk-free value of the swap is calculated by discounting the future net cash flows of the swap, based on the forward curve, using the forward yield curve for government instruments. This value is the same for all credit ratings. The probability of default from year 1 through until maturity can be either taken directly from Moody's or Standard & Poor's, or can be derived from the transition matrix (as we discussed earlier). The recovery rate is taken from statistical analyses provided by the rating agencies. Obviously, this ad hoc calculation of the exposure of an interest rate swap is not satisfactory. Only a model that makes use of stochastic interest rates will allow a proper treatment of exposure calculations for swaps and other derivative securities. 10 Conditional Transition Probabilities: CreditPortfolioView 19 CreditPortfolioView is a multifactor model that is used to simulate the joint conditional distribution of default and migration probabilities for various rating groups in different industries, and for

Page 345 each country, conditional on the value of macroeconomic factors. CreditPortfolioView is based on the observation that default probabilities and credit migration probabilities are linked to the economy. When the economy worsens, both downgrades and defaults increase; when the economy becomes stronger, the contrary holds true. In other words, credit cycles follow business cycles closely. Since the shape of the economy is, to a large extent, driven by macroeconomic factors, CreditPortfolioView proposes a methodology to link those macroeconomic factors to default and migration probabilities. It employs the values of macroeconomic factors such as the unemployment rate, the rate of growth in gross domestic product (GDP), the level of long-term interest rates, foreign exchange rates, government expenditures, and the aggregate savings rate. Provided that data are available, this methodology can be applied in each country to various sectors and various classes of obligors that react differently during the business cycle sectors such as construction, financial institutions, agriculture, services, etc. It applies better to speculative-grade obligors, whose default probabilities vary substantially with the credit cycle, than to investment-grade obligors, whose default probabilites are more stable. Default probabilities are modeled as a logit function, whereby the independent variable is a countryspecific index that depends upon current and lagged macroeconomic variables as described below: where Prob j,t is the conditional probability of default in period t, for speculative-grade obligors in country/industry j, and Y j,t is the country index value derived from a multifactor model described in Appendix 2. Note that the logit transformation (3) ensures that the probability takes a value between 0 and 1. In order to derive the conditional transition matrix, we employ the (unconditional Markov) transition matrix based on Moody's or Standard & Poor's historical data, which we denote by φm. Transition probabilities are unconditional in the sense that they are historical averages based on more than 20 years of data covering several business cycles, across many different countries and industries. As we discussed earlier, default probabilities for non-investment-grade obligors are higher than average during a period of recession. Also credit downgrades increase, while upward mi-

grations decrease. It is the opposite during a period of economic expansion. We can express this in the following way: Page 346 where SDP t is the simulated default probability for a speculative-grade obligor, based on equation (3), andφsdp t is the unconditional (historical average) probability of default for a speculative-grade obligor. CreditPortfolioView proposes to use these ratios (4) to adjust the migration probabilities in φm in order to produce a transition matrix, M, conditional on the state of the economy: where the adjustment consists in shifting the probability mass into downgraded and defaulted states when the ratio Prob j,t /φsdp is greater than 1, and vice versa if the ratio is less than 1. Since one can simulate Prob j,t over any time horizon t=1,..., T, this approach can generate multiperiod transition matrices: One can simulate the transition matrix (5) many times to generate the distribution of the cumulative conditional default probability for any rating, over any time period, as shown Figure 8.14. The same Monte Carlo methodology can be used to produce the conditional cumulative distributions of migration probabilities over any time horizon. KMV, described in Chapter 9, and CreditPortfolioView base their approach on the same empirical observation that default and migration probabilities vary over time. KMV adopts a microeconomic approach that relates the probability of default of any obligor to the market value of its assets. CreditPortfolioView proposes a methodology that links macroeconomic factors to default and migration probabilities. The calibration of CreditPortfolioView thus requires reliable default data for each country, and possibly for each industry sector within each country.

Page 347 Figure 8.14 Distribution of the Conditional Default Probability, for a Given Rating, over a Given Time Horizon, T Another limitation of the model is the ad hoc procedure to adjust the migration matrix. It is not clear that the proposed methodology performs better than a simple Bayesian model, where the revision of the transition probabilities would be based on the internal expertise accumulated by the credit department of the bank and an internal appreciation of the current stage of the credit cycle (given the quality of the bank's credit portfolio). These two approaches are somewhat related since the market value of the firm's assets depends on the shape of the economy; an interesting avenue of future research would be to compare the transition matrices produced by both models. Appendix 1 Elements of Merton's Model In this appendix we provide the basis of Merton's (1974) model of a firm's debt, as applied by CreditMetrics. Additional developments of the Merton model are presented in Chapter 9. The firm's assets value, V t, is assumed to follow a standard geometric Brownian motion, i.e:

with Z t ~ N(0,1), µ andσ 2 being, respectively, the mean and variance of the instantaneous rate of return on the assets of the firm, dv t /V t. V t is log-normally distributed with expected value at time t, standard Brownian motion, and variance equal to t. 20 Page 348 exp{µt}. The dynamics of V(t) are described by dv t /V t = µdt +σdw t, where W t is a is normally distributed with a zero mean and a It is assumed that the firm has a very simple capital structure, as it is financed only by equity, S t, and a single zero-coupon debt instrument maturing at time T, with face value F, and current market value B t (Table 8.9). The value of the assets of the firm is denoted by V t and: Denote by Prob(Def) the probability of default, i.e.: where V Def is the critical asset value below which default occurs. According to (1), default occurs when Z t satisfies: where the normalized return: is N[0,1]. Z CCC is simply the threshold point in the standard normal distribution, N(0,1), corresponding to a cumulative probabil-

ity of Prob Def. Then, the critical asset value V Def which triggers default is such that Z CCC = d 2 where: Page 349 and is also called "distance to default." 21 If we denote by r BB and r A the instantaneous rates of return on the assets of obligors that are rated BB and A, respectively, and byρthe instantaneous correlation coefficient between r A and r BB, then the normalized log-returns on both assets follow a joint normal distribution: This joint normal distribution is useful in calculating the jointmigration matrix for the two obligors initially rated A and BB. Consider two obligors whose probabilities of default are Prob(Def A ) and Prob(Def BB ), respectively. Their asset return correlation isρ. The events of default for obligors A and BB are denoted Def A and Def BB, respectively, and Prob(Def A, Def BB ) is the joint probability of default. Then, it can be shown that the default correlation is: 22 The joint probability of both obligors defaulting is, according to Merton's model: where V A and V BB denote the asset values for both obligors at time t, and V DefA and V DefBB are the corresponding critical values that trigger default. Expression (6) is equivalent to: where r A and r BB denote the normalized asset returns as defined

Page 350 in (3) for obligors A and BB, respectively, and are the corresponding distant to default as in (4). N 2 (x, y,ρ) denotes the cumulative standard bivariate normal distribution whereρis the correlation coefficient between x and y. Appendix 2 Default Prediction The Econometric Model Default probabilities are modeled as a logit function as follows: where Prob j,t is the conditional probability of default in period t, for speculative-grade obligors in country/industry j. Y j,t is the index value derived from a multifactor model described below. Note that the logit transformation ensures that the probability (1) takes a value between 0 and 1. The macroeconomic index, which captures the state of the economy in each country, is determined by the following multifactor model: where Y j,t is the index value in period t for the jth country/industry/speculative grade;β j,0,β j,1,β j,2,..., β j,m are coefficients to be estimated for the jth country/industry/speculative grade; X j,1,t, X j,2,t,..., X j,m,t are period t values of the macroeconomic variables for the jth country/industry; andν j,t is the error term assumed to be independent of X j,t and identically normally distributed. The macroeconomic variables are specified for each country. When sufficient data are available, the model can be calibrated at the country/industry level. Both the probability of default, P j,t, and the index, Y j,t, are then defined at the country/industry level, and the coefficient,β j,i are calibrated accordingly. In the proposed implementation, each macroeconomic variable is assumed to follow a univariate, autoregressive model of order 2 (AR2):

Page 351 where X j,i,t-1, X j,i,t-2 denote the lagged values of the macroeconomic variable X j,i,t ; γ j = (γ j,i,0, γ j,i,1 γ j,i,2 ) are coefficients to be estimated; e j,i,t is the error term assumed to be independent and identically distributed, i.e., e j,i,t ~ N(0,σ ej,i ) and e t ~ N(0,Σ e ), where e t denotes the vector of stacked error terms e j,i,t, of the j i AR(2) equations and Σe is the (j i) (j i) covariance matrix of the error terms e t. To calibrate the default probability model defined by (1), (2), and (3), one has to solve the system: where the vector of innovations E t is with where Σ ν,e and Σ e,ν denote the cross-correlation matrices. Once the system (4) has been calibrated, then one can use the Cholesky decomposition of Σ, i.e.: 23 to simulate the distribution of speculative default probabilities. First, draw a vector of random variables Z t, ~ N(0,1) where each component is normally distributed N(0,1). Then, calculate which is the stacked vector of error terms ν j,t and e j,i,t. Using these realizations of the error terms one can derive the corresponding values for Y j,t and Prob j,t.

Page 352 Appendix 3 Transition Matrix over a Period of Less Than One Year Given a one-year transition matrix, for example, the matrix given in Table 8.1, how can we derive the corresponding transition matrix over a period of time that is shorter than one year? For example, a period of six months, a quarter, a month, or 10 days as is required for the reporting of regulatory capital supporting the trading account? First, derive the eigen vectors x 1, x 2,... x n and the corresponding eigen valuesλ 1,λ 2,...,λ n of the one-year transition matrix T, where N denotes the number of credit categories, e.g., eight in Moody's eight-state rating system. The eigen values and vectors satisfy the property: Define X as the matrix of eigen vectors where the ith row is the transpose of x i, andλis a diagonal matrix where the ith diagonal element isλ i. A standard result in matrix algebra shows that: From this it is easy to see that the nth root of T, for example the 12th root for a monthly transition matrix, is: whereλ 1/n is the diagonal matrix where the ith diagonal element is. Notes 1. Specific risk refers to idiosyncratic or credit risk. It is the risk of an adverse price movement due to idiosyncratic factors related to the individual issuer of a security. 2. See also Vasicek (1997). 3. In the case of a loan commitment, a corporate borrower has the option to draw down on its credit line. It is more likely to exercise this option when its credit standing is deteriorating.

Page 353 4. CreditMetrics is a trademark of JP Morgan & Co., Inc. The technical document CreditMetrics (1997) provides a detailed exposition of the methodology, illustrated with numerical examples. CreditVaR is CIBC's proprietary credit value at risk model, which is part of CIBC's overall Credit Measurement Unit (CRMU) framework. The simple version implemented at CIBC, CreditVaR I, to capture specific risk for the trading book is based on the same principles as CreditMetrics. A more elaborate version, CreditVaR II, extends the CreditMetrics framework to allow for stochastic interest rates in order to address credit risk for derivatives including loan commitment and credit derivatives. 5. CreditMetrics' approach applies primarily to bonds and loans, which are both treated in the same manner. It can be easily extended to financial claims (such as receivables, financial letters of credit) for which we can derive the forward value at the risk horizon for all credit ratings. For derivatives such as swaps or forwards the model needs to be somewhat adjusted or ''twisted," since there is no satisfactory way to derive the exposure, and the loss distribution, within the proposed framework (since it assumes deterministic interest rates). 6. In Appendix 3 we show how to derive from the one-year transition matrix a transition matrix over a period less than one year, e.g., one month. 7. CreditMetrics calculates the forward value of the bonds, or loans, including compounded coupons paid out during the year. 8. See Carty and Lieberman (1996). See also Altman and Kishore (1996, 1998) for similar statistics. 9. The mean, m, and the variance,σ 2, of the distribution for V can be calculated from the data in Table 8.7 as follows:

Page 354 and σ = 2.99 The first percentile of a normal distribution N (m,σ 2 ) is m 2.33σ, i.e., 7.43. 10. See Appendix 1 for the derivation of the proof. In the next chapter we define the "distance to default" as the distance between the expected asset value and the default point. 11. Note that d 2 is different from its equivalent in the Black-Scholes formula since, here, we work with the "actual" instead of the "risk neutral" return distributions, so that the drift term in d 2 is the expected return on the firm's assets, instead of the risk-free interest rate as in Black-Scholes. See Chapter 1 for the definition of d 2 in Black-Scholes and Appendix 1 for d 2 in the above derivation. 12. The mathematical presentation of the joint probabilities and correlation of defaults is given in Appendix 1. 13. The correlation models for CreditMetrics and KMV are different. However, as the approaches are similar we detail only the more elaborate KMV model. 14. A good reference on Monte Carlo simulations and the Cholesky decomposition is Fishman (1997, p. 223). 15. If there were only one bond in the portfolio, PR would simply be the one-year spot rate on the corporate curve corresponding to the rating of the obligor. 16. See Chapter 14. 17. The RAROC concept will be presented and analyzed in Chapter 14. 18. See also KMV's correlation model, presented in the next chapter. 19. CreditPortfolioView is a risk measurement model developed by Tom Wilson (1997a, 1997b) and proposed by McKinsey & Company. 20. In fact the process for the market value of the firm should be written as where C denotes the net dollar payout by the firm to shareholders and bondholders. Here, we assume that no dividend or coupon is paid out debt is in the form of a zero coupon.

21. Note that d 2 is different from its equivalent in the Black-Scholes formula since, here, we work with the "actual" instead of the "risk neutral" return distributions, so that the drift term in d 2 is the expected return on the firm's assets, instead of the risk-free interest rate as in Black-Scholes. 22. See Lucas (1995). 23. See footnote 12. Page 355

Page 357 Chapter 9 The Contingent Claim Approach to Measuring Credit Risk 1 Introduction The CreditMetrics approach to measuring credit risk, as described in Chapter 8, is rather appealing as a methodology. Unfortunately it has a major weakness: reliance on ratings transition probabilities that are based on average historical frequencies of defaults and credit migration. As a result, the accuracy of CreditMetrics calculation depends upon two critical assumptions: first, that all firms within the same rating class have the same default rate and the same spread curve even when recovery rates differ among obligors; second, that the actual default rate is equal to the historical average default rate. The same assumptions also apply to the other transition probabilities. In other words, credit rating changes and credit quality changes are taken to be identical, and credit rating and default rates are also synonymous, i.e., the rating changes when the default rate is adjusted, and vice versa. This view has been strongly challenged (e.g., by researchers working for the consulting and software corporation KMV). 1 Indeed, these assumptions cannot be true since we know that default rates evolve continuously, while ratings are adjusted in a discrete fashion. (This lag is because rating agencies necessarily take time to upgrade or downgrade companies whose default risk has changed.)

Page 358 KMV has shown through a simulation exercise that the historical average default rate and transition probabilities can deviate significantly from the actual rates. In addition, KMV has demonstrated that substantial differences in default rates may exist within the same bond rating class, and that the overlap in default probability ranges may be quite large; for instance, some bonds rated BBB and AA may in fact exhibit the same probability of default. To illustrate this, KMV replicated 50,000 times, through a Monte Carlo simulation, Moody's study of default over a 25-year period. 2 For each rating they assumed a fixed number of obligors approximately the same number as in the Moody's study. For each rating they assumed that the true probability of default was equal to the reported Moody's average default rate over the 25-year period. KMV also ran the simulation for several levels of correlation among the asset returns, ranging from 15 percent to 45 percent. A typical result is illustrated in Figure 9.1 for a BBB obligor. Given Figure 9.1 Monte Carlo Simulated Distribution of Average Default Rate for a BBB Bond with a True Default Rate of 0.13 Percent Source: KMV Corporation.

Page 359 an exact default probability of 13 bp, the 25-year average historical default rate ranges between 4 bp and 27 bp at the 95 percent confidence level, for an asset correlation of 15 percent. The distribution is quite skewed, so that the mean or average default rate usually exceeds the typical (median) default rate for each credit class. Thus the average historical default probability overstates the default rate for a typical obligor. 3 What we will call the "structural" approach offers an alternative to the credit migration approach. Here, the economic value of default is presented as a put option on the value of the firm's assets. The merit of this approach is that each case can be analyzed individually based on its unique features. But this is also the principal drawback, since the information required for such an analysis is rarely available to the bank or the investor. Various ways to implement the structural approach have been proposed in the literature, all of which are consistent with arbitrage-free pricing methodologies. 4 The option pricing model approach, introduced by Merton (1974) in a seminal paper, 5 builds on the limited liability rule which allows shareholders to default on their obligations while they surrender the firm's assets to the various stakeholders, according to prespecified priority rules. The firm's liabilities are thus viewed as contingent claims issued against the firm's assets, with the payoffs to the various debt holders completely specified by seniority and safety covenants. Default occurs at debt maturity whenever the firm's asset value falls short of debt value at that time. In this model, the loss rate is endogenously determined and depends on the firm's asset value, volatility, and the default-free interest rate for the debt maturity. 6 An alternative to this approach, proposed by Longstaff and Schwartz (1995) and described in Appendix 3 of this chapter, allows bankruptcy to occur at a random default time. Bankruptcy is triggered the first time the value of the firm's assets falls to some prespecified default boundary; the approach also assumes that the loss in the event of default is exogenous. This approach simplifies the modeling of the bankruptcy process by not relying explicitly on the priority structure of the debt instruments. However, it loses its generality by assuming an exogenous recovery rate for each promised dollar in the case of a default. These models allow for stochastic interest rates.

Page 360 The most recent reduced-form approach, developed independently by Duffie and Singleton (1994), Jarrow and Turnbull (1995), and Jarrow, Lando, and Turnbull (1997) and summarized in Chapter 10, characterizes bankruptcy as an exogenous process, e.g., as a Markov process in the firm's credit ratings. Thus the approach does not explicitly depend on the firm's asset value, and on the priority rules for the various debt instruments. However, the approach still assumes a given recovery rate in the event of default. Contrary to the earlier approaches, the default event does not relate to the capital structure of the firm and occurs at a random time. 7 These models allow one to derive the term structure of default probabilities from credit spreads, while assuming exogenous and somewhat arbitrary the recovery rate. 8 In this chapter we adopt the "traditional" option pricing framework to value corporate securities, and we show that it allows us to repeat results derived by Jarrow and Turnbull (1995). That is, the credit spread on a corporate bond is the product of the probability of default and the loss rate. In Section 2 we present the economic value of default as a put option. A numerical example is used to illustrate the application of option pricing theory to the assessment of credit risk. The probability of default and the conditional expected recovery rates are derived from the model described in Section 3. 2 A Structural Model of Default Risk: Merton's (1974) Model 9 The model presented in this section assumes a simple capital structure with one type of (zero-coupon) debt. It can, however, be easily extended to the case where the firm has issued senior and junior debt. In this case, the loss rates for each type of debt are endogenously derived, together with the default probability. 10 Consider the simple case of a firm with risky assets V, which are financed by equity S and by one debt obligation, maturing at time T with face value (including accrued interest) of F and market value B. The loan to the firm is subject to credit risk, namely the risk that at time T the value of the firm's assets, V T, will fall below the obligation to the debt holders, F. Credit risk exists as long as the probability of default, Prob (V T < F), is greater than zero. This implies that at time 0, B 0 < Fe rt ; i.e., the yield to maturity on the debt, y T, is higher than the

Page 361 risk-free rate r, where π T = y T r denotes the default spread that compensates the bond holders for the default risk that they bear. If we assume that markets are frictionless, with no taxes, and there is no bankruptcy cost, then the value of the firm's assets is simply the sum of the firm's equity and debt, i.e.: From the viewpoint of a bank that makes a loan to the firm, this gives rise to a series of questions. Can the bank eliminate/reduce credit risk, and at what price? What is the economic cost of reducing credit risk? And, what are the factors affecting this cost? In this simple framework, credit risk is a function of the financial structure of the firm, i.e., its leverage ratio LR Fe rt /V 0 (where V 0 is the present value of the firm's assets, and Fe rt is the present value of the debt obligation at maturity, assuming debt is riskless), the volatility of the rate of return of the firm's assets, σ, and the time to maturity of the debt, T. The model was initially suggested by Merton (1974) and further analyzed by Galai and Masulis (1976). To determine the value of the credit risk arising from this bank loan, we first make two assumptions: that the loan is the only debt instrument of the firm, and that the only other source of financing is equity. In this case, the credit value is equal to the value of a put option on the value of the assets of the firm, V, at a strike price of F, maturing at time T. If the bank purchased such a put option, it would completely eliminate the credit risk associated with the loan (see Table 9.1). To phrase this another way, by purchasing the put on V for the term of the debt, with a strike price equal to the face value of TABLE 9.1 Bank's Pay-Off Matrix At Times 0 and T for Making a Loan and Buying a put Option Time 0 T Value of assets V 0 V T F V T >F Bank's position: (a) Make a loan B 0 V T F (b) Buy a put P 0 F V T 0 Total B 0 P 0 F F

the loan, the bank can completely eliminate all the credit risk and convert the risky corporate loan into a riskless loan with a face value of F. If the riskless interest rate is r, then in equilibrium it should be that B 0 + P 0 = Fe rt. Page 362 Thus, the value of the put option is the cost of eliminating the credit risk associated with providing a loan to the firm. * If we make the assumptions that are needed to apply the Black Scholes (1973) (BS) model to equity and debt instruments (see Galai and Masulis 1976 for a detailed discussion of the assumptions), we can write the value of the put as: where P 0 is the current value of the put, N(.) is the cumulative standard normal distribution, and andσis the standard deviation of the rate of return of the firm's assets. The model illustrates that the credit risk, and its costs, is a function of the riskiness of the assets of the firm,σ, and this risk is also a function of the time interval until debt is paid back, T. The cost is also affected by the risk-free interest rate r: the higher r is, the less costly it is to reduce credit risk. The cost is a homogeneous function of the leverage ratio, LR = Fe rt /V 0, which means that it stays constant for a scale expansion of Fe rt /V 0. We can now derive the yield to maturity for the corporate discount debt, y T, as follows: so that the default spread,π T, defined asπ T = y T r, can be derived from equation (2): * This put is an example of a credit derivative (see Chapter 12).

Page 363 The default spread can be computed exactly as a function of the leverage ratio, LR Fe rt /V 0, the volatility of the underlying assets, σ, and the debt maturity, T. The numerical examples in Table 9.2 show the default spread for various levels of volatility and different leverage ratios. Note that when the risk-free rate r increases, the credit spread π T declines, i.e., π T / r < 0. Indeed, the higher r is, the less risky is the bond (the lower is the value of the put protection). Therefore, the lower the risk premium π T. In Table 9.2, by using the BS model when V 0 = 100, T = 1, r = 10 percent, and also σ = 40 percent with the leverage ratio (LR) = 70 percent, 11 we obtain for the value of equity, S 0 = 33.37 and the value of the corporate risky debt, B 0 = 66.63. The yield on the loan is equal to 77/66.63 1 = 0.156; i.e., there is a 5.6 percent risk premium on the loan to reflect the credit risk. The model also shows that the put value is P 0 = 3.37. Hence the cost of eliminating the credit risk is $3.37 for $100 worth of the firm's assets, where the face value (i.e., the principal amount plus the promised interest rate) of the one-year debt is 77. This cost drops to 25 cents when volatility decreases to 20 percent, and to 0 for 10 percent volatility. The riskiness of the assets as measured by the volatility σ is a critical factor in determining credit risk. TABLE 9.2 Default Spread For Corporate Debt (For V 0 = 100, T = 1, and r = 10% 1 ) Leverage ratio: LR Volatility of underlying asset: σ 0.05 0.10 0.20 0.40 0.5 0 0 0 1.0% 0.6 0 0 0.1% 2.5% 0.7 0 0 0.4% 5.6% 0.8 0 0.1% 1.5% 8.4% 0.9 0.1% 0.8% 4.1% 12.5% 1.0 2.1% 3.1% 8.3% 17.3% 1 10% is the annualized interest rate discretely compounded, which is equivalent to 9.5% continuously compounded.

To demonstrate that the bank eliminates all its credit risk by buying the put, we can compute the yield on the bank's position as: Page 364 which translates to a riskless yield of 10 percent per annum. In Appendix 1 we show how the conventional analysis, based on yield spreads, can be transformed into the options approach. 3 Probability of Default, Conditional Expected Recovery Value, and Default Spread From equation (2) one can extract the probability of default for the loan. In a risk-neutral world, N(d 2 ) is the probability that the firm's value at time T will be higher than F, and 1 N(d 2 ) = N( d 2 ) is the probability of default. By purchasing the put P 0, the bank buys an insurance policy whose premium is the discounted expected value of the expected shortfall in the event of default. Indeed, equation (2) can be rewritten as: Equation (4) decomposes the premium on the put into three factors. The absolute value of the first term inside the bracket is the expected discounted recovery value of the loan, conditional on V T F. It represents the risk-neutral expected payment to the bank in the case where the firm is unable to pay the full obligation F at time T. The second term in the bracket is the current value of a riskless bond promising a payment of F at time T. Hence, the sum of the two terms inside the brackets yields the expected shortfall in presentvalue terms, conditional on the firm being bankrupt at time T. The final factor which determines P 0 is the probability of default, N( d 2 ). By multiplying the probability of default by the current value of the expected shortfall we derive the premium for insurance against default. Using the same numerical example as in the previous section (i.e., V 0 = 100, T = 1, r = 10 percent,σ = 40 percent, F = 77, and LR = 0.7), we obtain:

Page 365 The above results are based on an assumption of risk neutrality. For the general case, when the assumption of a risk-neutral world is removed, the probability of default is given by where: 12 and where µ is the expected rate of return on asset V, and V is assumed to be log-normally distributed. See Boness (1964) and Galai (1978) for an explanation of this result. Referring to our numerical example, the risk-neutral probability of default is 24.4 percent. If we assume that the discrete time µ is 16 percent, then the probability of default is 20.5 percent. The expected recovery value is now: From (4) we can compute the expected loss EL T in the event of default, at maturity date T: Again, using our previous numerical example, we obtain: This result is consistent with the definition of the default spread and its derivation in (3). Indeed, from (5) the expected payoff from the corporate debt at maturity is:

Page 366 so the expected cost of default, expressed in yield, is: which is identical to (3). The result in equation (5) is similar to the conclusion in Jarrow and Turnbull's (1995) model, which is used to price credit derivatives; i.e., the credit spread is the product of the probability of default and the loss in the event of default. However, in their model they assume that the term structure of credit spread is known and can be derived from market data. The forward spread can then be easily derived. By assuming that the recovery factor is given and exogenous to the model, they can imply the forward probability of default (see also Appendix 1 and Chapter 10). In the contingent claim model we reach the same conclusion, but both the probability of default and the recovery rate are simultaneously derived from equilibrium conditions. From equations (3) and (4) it is clear that the recovery rate cannot be assumed to be constant: it varies as a function of time to maturity, and according to the value of the firm's assets. 4 Estimating Credit Risk as a Function of Equity Value In Section 2 we showed that the cost of eliminating credit risk can be derived from the value of the firm's assets. A practical problem arises over how easy it is to observe V. In some cases, if both equity and debt are traded, V can be reconstructed by adding the market values of both equity and debt. However, corporate loans are not often traded and so, to all intents and purposes, we can only observe equity. The question, then, is whether the risk of default can be hedged by trading shares and derivatives on the firm's stock. In our simple framework, equity itself is a contingent claim on the firm's assets. Its value can be expressed as:

Page 367 Equity value is a function of the same parameters as the put calculated in equation (2). A put can be created synthetically by selling short N( d 1 ) units of the firm's assets, and buying Fe rt N( d 2 ) units of government bonds maturing at T, with face value of F. If one sells short N( d 1 )/ N(d 1 ) units of the stock S, one effectively creates a short position in the firm's assets of N( d 1 ) units, since: Therefore, if V is not directly traded or observed, one can create a put option dynamically by selling short the appropriate number of shares. The equivalence between the put and the synthetic put is only valid over short time intervals, and must be readjusted frequently with changes in S and in time left to debt maturity. Using the data from the previous numerical example, N( d 1 ) / N(d 1 ) = 0.137 / 0.863 = 0.159. This means that in order to insure against the default of a one-year loan with a maturity value of 77, for a firm with a current market value of assets of 100, the bank should sell short 0.159 of the outstanding equity. [Note that the outstanding equity is equivalent to a short-term holding of N(d 1 ) = 0.863 of the firm's assets. Shorting 0.159 of equity is equivalent to shorting 0.863 of the firm's assets.] The question now is whether we can use a put option on equity in order to hedge the default risk. It should be remembered that equity itself reflects the default risk, and as a contingent claim its instantaneous volatilityσ s can be expressed as: whereη S,V = N(d 1 )V / S is the instantaneous elasticity of equity with respect to the firm's value, andη S,V 1. Sinceσ s is stochastic, changing with V, the conventional BS model cannot be applied to the valuation of puts and calls on S. The BS model requires σ to be constant, or to follow a deterministic path over the life of the option. However, it was shown by Bensoussan, Crouhy, and Galai (1994, 1995) that a good approximation can be achieved by employing equation (7) in the BS model. In practice, for long-term options, the estimatedσ s from (7) is not expected to change widely from day to day. Therefore, equation

(7) can be used in the context of BS estimation of long-term options, even when the underlying instrument does not follow a stationary log-normal distribution. 5 KMV Approach Page 368 KMV derives the estimated default frequency or default probability, the EDF, for each obligor based on the Merton (1974) type of model. The probability of default is thus a function of the firm's capital structure, the volatility of the asset returns, and the current asset value. 13 The EDF is firm specific, and can be mapped onto any rating system to derive the equivalent rating of the obligor. EDFs can be viewed as a ''cardinal ranking" of obligors relative to default risk, instead of the more conventional "ordinal ranking" proposed by rating agencies (which relies on letters such as AAA, AA, etc). Contrary to CreditMetrics, KMV's model does not make any explicit reference to the transition probabilities which, in KMV's methodology, are already embedded in the EDFs. Indeed, each value of the EDF is associated with a spread curve and an implied credit rating. Credit risk in the KMV approach is essentially driven by the dynamics of the asset value of the issuer. Given the capital structure of the firm, 14 and once the stochastic process for the asset value has been specified, then the actual probability of default for any time horizon, one year, two years, etc., can be derived. Figure 9.2 schematizes how the probability of default relates to the distribution of asset returns and the capital structure of the firm. We assume that the firm has a very simple capital structure. It is financed by means of equity S t and a single zero-coupon debt instrument maturing at time T, with face value F, and current market value B t. The firm's balance sheet can be represented as follows: V t B t (F) + S t, where V t is the value of all the assets. The firm's assets value V t is assumed to follow a standard geometric Brownian motion (see Chapter 8). In this framework, default only occurs at maturity of the debt obligation, when the value of assets is less than the promised payment F to the bond holders. Figure 9.2 shows the distribution of the assets' value at time T, the maturity of the zero-coupon debt, and the probability of default (i.e., the shaded area below F).

The KMV approach is best applied to publicly traded companies, where the value of the equity is determined by the stock market. The information contained in the firm's stock price and balance sheet can then be translated into an implied risk of default as shown in the next subsections. The derivation of the actual probabilities of default proceeds in three stages: Estimation of the market value and volatility of the firm's assets Calculation of the distance to default, which is an index measure of default risk Scaling of the distance to default to actual probabilities of default using a default database Page 369 Figure 9.2 Distribution of the Firm's Asset Value at Maturity of the Debt Obligation

Page 370 5.1 Estimation of the Asset Value, V, and the Volatility of Asset Return, σ In the contingent claim approach to the pricing of corporate securities, the market value of the firm's assets is assumed to be log-normally distributed; i.e., the log-asset return follows a normal distribution. 15 This assumption is quite robust and, according to KMV's own empirical studies, actual data conform quite well to this hypothesis. 16 In addition the distribution of asset returns is stable over time; i.e., the volatility of asset returns remains relatively constant. As we discussed earlier, if all the liabilities of the firm were traded, and marked-to-market every day, then the task of assessing the market value of the firm's assets and its volatility would be straightforward. The firm's asset value would be simply the sum of the market values of the firm's liabilities, and the volatility of the asset return could be simply derived from the historical time series of the reconstituted asset value. In practice, however, only the price of equity for most public firms is directly observable, and in some cases only part of the debt is actively traded. The alternative approach to asset valuation consists of applying the option pricing model to the valuation of corporate liabilities as suggested in Merton (1974). 17 In order to make their model tractable, KMV assumes that the capital structure of a corporation is composed solely of equity, short-term debt (considered equivalent to cash), long-term debt (in perpetuity), and convertible preferred shares. 18 Given these simplifying assumptions, it is possible to derive analytical solutions for the value of equity, S, and its volatility,σ S : where LR denotes the leverage ratio in the capital structure, c is the average coupon paid on the longterm debt, and r is the risk-free interest rate. Ifσ S were directly observable, like the stock price, we could solve simultaneously (8) and (9) for V andσ. But the instantaneous equity volatility,σ S, is relatively unstable, and is in fact quite sensitive to the change in asset value; there is no simple way to measure

Page 371 preciselyσ S from market data. 19 Since only the value of equity, S, is directly observable, we can back out V from (8) so that it becomes a function of the observed equity value, or stock price, and the volatility of asset returns: To calibrate the model forσ, KMV uses an iterative technique. 5.2 Calculation of the Distance to Default In the option pricing framework default, or equivalently bankruptcy, occurs when the asset value falls below the value of the firm's liabilities. In practice, default is distinct from bankruptcy. Bankruptcy describes the situation in which the firm is liquidated, and the proceeds from the asset sale are distributed to the various claim holders according to prespecified priority rules. Default, on the other hand, is usually defined as the event when a firm misses a payment on a coupon and/or the reimbursement of principal at debt maturity. Cross-default clauses on debt contracts are such that when the firm misses a single payment on a debt, it is declared in default on all its obligations. Since the early 1980s, Chapter 11 regulation in the United States has protected firms in default and helped to maintain them as going concerns during a period in which they attempt to restructure their activities and their financial structure. Figure 9.3 compares the number of bankruptcies to the number of defaults during the period 1973 to 1994. Using a sample of several hundred companies, KMV observed that firms default when the asset value reaches a level that is somewhere between the value of total liabilities and the value of short-term debt. Therefore, the tail of the distribution of asset values below total debt value may not be an accurate measure of the actual probability of default. Loss of accuracy may also result from factors such as the non-normality of the asset return distribution, and the simplifying assumptions about the capital structure of the firm. This may be further aggravated if a company is able to draw on (otherwise unobservable) lines of credit. If the company is in distress, using these lines may (unexpectedly) increase its liabilities while providing the necessary cash to honor promised payments.

Figure 9.3 Bankruptcies and Defaults, Quarterly from 1973 1997 Source: KMV Corporation. Page 372

Page 373 Figure 9.4 Distance to Default (DD) STD = LTD = DPT = DD = short-term debt long-term debt default point = STD + 1/2 LTD distance to default, which is the distance between the expected asset value in one year, E(V 1 ), and the default point, DPT. It is often expressed in terms of standard deviation of asset returns: For all these reasons, KMV implements an intermediate phase before computing the probabilities of default. As shown in Figure 9.4, which is similar to Figure 9.2, KMV computes an index called "distance to default" (DD). DD is the number of standard deviations between the mean of the distribution of the asset value, and a critical threshold, the "default point," set at the par value of current liabilities including short-term debt to be serviced over the time horizon, plus half the longterm debt.

Page 374 Given the log-normality assumption of asset values, the distance to default, expressed in units of asset return standard deviation at time horizon T, is: 20 where V 0 = current market value of assets DPT T = default point at time horizon T µ = expected return on assets, net of cash outflows σ = annualized asset volatility It follows that the shaded area below the default point is equal to N( DD). 5.3 Derivation of the Probabilities of Default from the Distance to Default This last phase consists of mapping the "distance-to-default" (DD) to the actual probabilities of default, for a given time horizon (see Figure 9.5). KMV calls these probabilities "expected default frequencies," or EDFs. Using historical information about a large sample of firms, including firms that have defaulted, one can estimate, for each time horizon, the proportion of firms of a given ranking, say DD = 4, that actually defaulted after one year. This proportion, say 40 bp, or 0.4 percent, is the EDF as shown in Figure 9.5. Example 1 Current market value of assets: V 0 = 1000 Net expected growth of assets per annum: 20% Expected asset value in one year: V 0 (1.20) = 1200 Annualized asset volatility, σ: 100 Default point: 800 Then Assume that among the population of all the firms with a DD of 4 at one point in time, e.g., 5000, 20 defaulted one year later; then:

Page 375 The implied rating for this probability of default is BB +. The next example is provided by KMV and relates to Federal Express on two different dates: November 1997 and February 1998. Example 2: Federal Express ($ figures are in billions of $US) Market capitalization (price shares outstanding) November 1997 February 1998 $ 7.7 $ 7.3 Book liabilities $ 4.7 $ 4.9 Market value of assets $12.6 $12.2 Asset volatility 15% 17% Default point $ 3.4 $ 3.5 Distance to default (DD) EDF 0.06% (6 bp) AA 0.11% (11 bp) A Figure 9.5 Mapping of the "Distance to Default" into the EDFs for a Given Time Horizon

Page 376 This last example illustrates the main causes of change for an EDF, i.e., variations in the stock price, the debt level (leverage ratio), and the asset volatility (i.e., the perceived degree of uncertainty concerning the value of the business). 5.4 EDF as a Predictor of Default KMV has provided a "Credit Monitor" service for estimated EDFs since 1993. EDFs have proved to be a useful leading indicator of default, or at least of the degradation of the creditworthiness of issuers. When the financial situation of a company starts to deteriorate, EDFs tend to shoot up quickly until default occurs, as shown in Figure 9.6. Figure 9.7 shows the evolution of equity value and asset value, as well as the default point during the same period. On the vertical axis of both graphs the EDF is shown as a percentage, together with the corresponding Standard & Poor's rating. KMV has analyzed more than 2000 U.S. companies that have defaulted or entered into bankruptcy over the last 20 years. These firms belonged to a large sample of more than 100,000 company-years with data, provided by Compustat. In all cases KMV has shown a sharp increase in the slope of the EDF a year or two prior to default. Changes in EDFs tend to anticipate by at least one year the downgrading of the issuer by rating agencies such as Moody's and Standard & Poor's (Figure 9.8). Contrary to Moody's and Standard & Poor's historical default statistics, EDFs are not biased by periods of high or low defaults. The distance to default can be observed to shorten during periods of recession, when default rates are high, and to increase during periods of prosperity characterized by low default rates. 5.5 EDFs and Ratings Standard & Poor's risk ratings represent default probabilities only, while Moody's factors also include a measure of the probability of loss, i.e., EDF LGD. Table 9.3 shows the correspondence between EDFs and the ratings systems used by Standard & Poor's and Moody's, as well as the internal risk ratings systems used by CIBC, Nationsbank, and Swiss Bank Corp. (The ratings systems of

Page 377 Figure 9.6 EDF of a Firm That Defaulted versus EDFs of Firms in Various Quartiles and the Lower Decile. Note: The quartiles and decile represent a range of EDFs for a specific credit class (B rated firms). Source: KMV Corporation. Nationsbank and Swiss Bank were published during recent CLO transactions.) Within any rating class the default probabilities of issuers are clustered around the median. However, as we discussed earlier, the average default rate for each class is considerably higher than the default rate of the typical firm. This is because each rating class contains a group of firms which have much higher probabilities of default, due to the (approximately) exponential change in default rates as default risk increases. These risky firms can be thought of as firms that should have been downgraded, though no downgrade has yet occurred. Conversely, there are also firms within each class

Page 378 Figure 9.7 Asset Value, Equity Value, Short-Term Debt, and Long-Term Debt of a Firm That Defaulted Source: KMV Corporation. that should have been upgraded. Table 9.4 shows the variation of the EDFs within each rating class. Three consequences follow from this analysis. First, since the rating agencies are slow to change their ratings, the historical frequency of remaining in a rating class should overstate the true probability of maintaining a particular credit quality. Second, the average historical probability of default overstates the true probability of default for typical firms within each rating class, due to the difference between the mean and the median default rates. Third, if both the probability of staying in a given rating class and the probability of default are too large, then the transition probabilities must be too small. KMV has constructed a transition matrix based upon default rates rather than rating classes. They began by ranking firms into

Page 379 Figure 9.8 EDF of a Firm That Defaulted versus Standard & Poor's Rating Source: KMV Corporation. TABLE 9.3 EDFs and Risk Rating Comparisons EDF S&P Moody's Factors CIBC Nationsbank SBC 2 to 4 bp >=AA >=Aa2 1 AAA C1 4 to 10 bp AA/A A1 2 AA C2 10 to 19 bp A/BBB+ Baa1 3 A C3 19 to 40 bp BBB+/BBB- Baa3 4 A/BB C4 40 to 72 bp BBB-/BB Ba1 5 BBB/BB C5 72 to 101 bp BB/BB- Ba3 6 BB C6 101 to 143 bp BB-/B+ B1 7 BB C7 143 to 202 bp B+/B B2 8 BB/B C8 202 to 345 bp B/B- B2 9 B C9

Page 380 TABLE 9.4 Variation of EDFs Within Rating Quantiles 10 25 50 75 90 Mean AAA 0.02 0.02 0.02 0.02 0.10 0.04 AA 0.02 0.02 0.02 0.04 0.10 0.06 A 0.02 0.03 0.08 0.13 0.28 0.14 BBB 0.05 0.09 0.15 0.33 0.71 0.30 BB 0.12 0.22 0.62 1.30 2.53 1.09 B 0.44 0.87 2.15 3.80 7.11 3.30 CCC 1.43 2.09 4.07 12.24 18.82 7.21 Source: KMV Corporation. groups that were based on the nonoverlapping ranges of default probabilities that are typical for a rating class. For instance, all firms with an EDF of less than 2 bp were ranked AAA; those with an EDF of between 3 bp and 6 bp were placed in the AA group; firms with an EDF of 7 bp to 15 bp were placed in the A rating class; and so on. Then, using the history of changes in EDFs, the transition TABLE 9.5 KMV One-Year Transition Matrix Based on Nonoverlapping EDF Ranges Initial Rating Rating at Year-End (%) AAA AA A BBB BB B CCC Default AAA 66.26 22.22 7.37 2.45 0.86 0.67 0.14 0.02 AA 21.66 43.04 25.83 6.56 1.99 0.68 0.20 0.04 A 2.76 20.34 44.19 22.94 7.42 1.97 0.28 0.10 BBB 0.30 2.80 22.63 42.54 23.52 6.95 1.00 0.26 BB 0.08 0.24 3.69 22.93 44.41 24.53 3.41 0.71 B 0.01 0.05 0.39 3.48 20.47 53.00 20.58 2.01 CCC 0.00 0.01 0.09 0.26 1.79 17.77 69.94 10.13 Source: KMV Corporation.

Page 381 TABLE 9.6 Transition Matrix Based on Actual Rating Changes Initial Rating Rating at Year-End (%) AAA AA A BBB BB B CCC Default AAA 90.81 8.33 0.68 0.06 0.12 0 0 0 AA 0.70 90.65 7.79 0.64 0.06 0.14 0.02 0 A 0.09 2.27 91.05 5.52 0.74 0.26 0.01 0.06 BBB 0.02 0.33 5.95 86.93 5.30 1.17 1.12 0.18 BB 0.03 0.14 0.67 7.73 80.53 8.84 1.00 1.06 B 0 0.11 0.24 0.43 6.48 83.46 4.07 5.20 CCC 0.22 0 0.22 1.30 2.38 11.24 64.86 19.79 Source: Standard & Poor's CreditWeek (April 15, 1996). matrix shown in Table 9.5 was produced. It is similar in structure to the table produced in Table 8.1 of Chapter 8 and reproduced here as Table 9.6. There are striking, if predictable, differences in the various probabilities between the two tables. According to KMV, except for the AAA rating class, the probability of staying in the same rating class is between half and one-third of historical rates produced by the rating agencies. KMV's probabilities of default are also lower, especially for the low-grade credits. Migration probabilities are also much higher as calculated by KMV, especially for the grade above and below the current rating class. These differences may have a considerable impact on credit VaR calculations. 6 KMV's Valuation Model for Cash Flows Subject to Default Risk In the CreditMetrics approach, the valuation model is quite simplistic. As we described in Chapter 8, if the time horizon is one year, then the forward value of a bond is the discounted value of the future cash flows beyond one year, where the discount factors are derived from the forward yield curve. Each credit rating is

Page 382 associated with a specific spread curve, and the distribution of future values follows from the transition probabilities. The KMV approach is quite different, and is consistent with the option pricing methodology used for the valuation of contingent cash flows. Given the term structure of EDFs for a given obligor, we can derive the net present value of any stream of contingent cash flows. More specifically, KMV's pricing model is based upon the ''risk-neutral" valuation model, also known as the "martingale" approach to the pricing of securities. This derives prices as the discounted expected value of future cash flows using so-called "risk-neutral" probabilities (i.e., not the actual probabilities as they can be observed in the marketplace from historical data or from the EDFs). 21 Assuming, for the time being, that we know how to derive the "risk-neutral probabilities" from the EDFs, then the valuation of risky cash flows proceeds in two steps: first, the valuation of the defaultfree component; and second, the valuation of the component that is exposed to credit risk. In Appendix 2 we illustrate how to use risk-neutral probabilities to value a bond or a loan subject to default risk, and how to use them to value a credit derivative. 6.1 Derivation of the "Risk-Neutral" EDFs Equation (5) in Section 3 illustrates how the risk-neutral probability of default is used in order to estimate the expected loss in present-value terms from a risky zero-coupon bond. A similar procedure can be used for a multiperiod bond with a given coupon payment. The practical problem faced by the analyst is that the empirical frequencies of default deviate from the risk-neutral probabilities. In theory, the difference for the simple case discussed in Section 3 is between N( d 2 ) for risk-neutral probability and N( ) for actual probability of default (24.4 percent and 20.5 percent, respectively, in our numerical example). While N( d 2 ) is a function of r, the riskfree interest rate, N( ) is a function of µ, the expected rate of return on the assets of the firm (and µ > r). Since, it follows that the cumulative risk-neutral EDF, Q T, at horizon T can be expressed as a function of the EDF:

Page 383 Since µ r, it follows that Q T EDF T ; i.e., the risk-neutral probability of default, after adjusting for the price of risk, is higher than the actual probability of default. According to the continuous time CAPM: with where r V and r M denote the continuous time rate of return on the firm's asset and the market portfolio, respectively;σ andσ M are the volatility of the assets return and the market return, respectively;ρ V, M is the correlation between the assets return and the market return; and where µ M denotes the expected return on the market portfolio, and r is the continuous time risk-free rate. It follows that: where SR =π/σ M denotes the market Sharpe ratio, i.e., the excess return per unit of market volatility for the market portfolio. Substituting (16) into (12) we obtain: ρ V, M is estimated in terms of the linear regression of asset returns against market returns: whereαis the intercept of the regression andεis the error term.ρ V, M is simply the square root of the R-squared of this regression.

Page 384 In practice,π, the market risk premium, is difficult to estimate statistically, and it varies over time. In addition, the EDF is not precisely the shaded area under the default point in Figure 9.2, and the asset return distribution is not exactly normal. For all these reasons, KMV estimates the risk-neutral EDF, Q T, by calibrating the market Sharpe ratio, SR, andθin the following relation, using bond data: whereθ is a time parameter which should, in theory, be equal to 1/2. Assuming we have derived the zero-coupon curve for an obligor, then according to the pricing model (A3) presented in Appendix 2: for i = 1,..., n, where: r v,i = continuously compounded zero-interest rate for the obligor, for maturity t i, r i = continuously compounded zero-risk free rate, for maturity t i so that: Combining (15) and (17) we obtain: where r v,i r i is the obligor's corporate spread for maturity t i, which is directly extracted from corporate bond data. 22 SR andθare calibrated to produce the best fit of (22) in the least-square sense. 7 Asset Return Correlation Model CreditMetrics and KMV derive asset return correlations from a structural model that links correlations to fundamental factors. By imposing a structure on the return correlations, sampling errors inherent in simple historical correlations are avoided, and a better accuracy in forecasting correlations is achieved. In addition, there is a practical need to reduce dramatically the number of correlations that need to be calculated. Assume that

Page 385 a bank is dealing with N = 1000 different counterparties. Then, we have N(N 1)/2 different correlations to estimate, i.e., 499,500. This is a staggering number. Multifactor models of asset returns reduce the number of correlations that have to be calculated to the limited number of correlations between the common factors that affect asset returns. It is assumed that the firm's asset returns are generated by a set of common, or systematic, risk factors and idiosyncratic factors. The idiosyncratic factors are either firm specific, or country or industry specific, and do not contribute to asset return correlations (since they are not correlated with each other, and not correlated with the common factors). Asset return correlations between two firms can be explained only in terms of the factors common to all firms. Only the risks associated with the idiosyncratic risk factors can be diversified away through portfolio diversification, while the risk contribution of the common factors is, on the contrary, nondiversifiable. For the sake of illustration, assume the asset return generating process for all firms is: where: N = number of obligors (firms) r k = asset return for firm k (r vk denoted r k ) α k = component of asset return independent of common factors I 1,I 2 = common factors β 1k,β 2k = expected changes in r k, given a change in common factors I 1 and I 2, respectively ε k = idiosyncratic risk factor with zero mean, and assumed to be uncorrelated with all the common factors, as well as with the idiosyncratic risk factors of the other firms Then, using elementary statistical calculus, we can derive results that are well known in portfolio theory: 23

Page 386 If we denote byρ ij the asset return correlation between firm i and firm j, then: To derive the asset return correlation between any number of firms, we only need, according to (24 26), to estimate theβ ik 's for i = 1, 2, and K = 1,..., N, i.e., 2N parameters, and the covariance matrix for the common factors, i.e., three parameters. In the previous example, where we considered N = 1000 firms, the implementation of this two-factor model would require us to estimate only 2003 parameters (instead of 499,500 different historical asset return correlations). For K common factors, the number of parameters to be estimated is KN + K(K - 1)/2. If K = 10, then this number becomes 10,045. This result can be easily generalized to any number of common factors and idiosyncratic risk components. The next problem is to specify the structure of the factors. CreditMetrics and KMV propose relatively similar models, so here we will present only the KMV model (which is more comprehensive and elaborate). 24 KMV constructs a three-layer factor structure model as shown in Figure 9.9: First level: a composite company-specific factor, which is constructed individually for each firm based on the firm's exposure to each country and industry Second level: country and industry factors Third level: global, regional, and industrial sector factors The first level of the structure divides between firm-specific, or idiosyncratic, risk and common, or systematic, risk. Systematic risk is captured by a single composite index, which is firm specific, and which is constructed as a weighted sum of the firm's exposure to country and industry factors defined at the second level of the structure:

Page 387 where r k = asset return for firm k CF k = composite factor for firm k β k = firm k's response to the composite factor, i.e., expected change in r k given a change in the composite factor ε k = firm k's specific risk factor The composite factor is constructed as the sum of the weighted country and industry factors specified at the second level of the structure: Figure 9.9 Factor Model for Asset Return Correlations Source: KMV Corporation.

Page 388 where: C m = rate of return on country risk factor m I n = rate of return on industry risk factor n α km = weight of firm k in country m, with the constraint thatσ m α km = 1 α kn = weight of firm k in industry n, with the constraint thatσ n α kn = 1 For example, consider a Canadian firm that has two lines of business, and assume that the data are extracted from Compustat: 25 Business line SIC 26 Assets Sales Lumber and forestry 2431 35% 45% Paper production 2611 65% 55% Total 100% 100% To determine the weight by industry, we average the asset and sales breakdowns. Thus for the above example the weight for lumber and forestry is: and for paper it is: Note that, by construction, the weights add up to 100 percent. The country exposures are calculated in a similar manner and should also sum to 100 percent. (In this example, we assume a 100 percent exposure to Canada.) Then the composite factor can be written as: At the third level of the factor structure, the risk of countries and industries is further decomposed into systematic and idiosyncratic components. The systematic component is captured by basic factors such as global economic effect, regional factor effect, and

Page 389 sector factor effect. While the common factor is firm specific, the third-level factors are the same for all countries and all industries: We can now express this factor structure in a form similar to (23), from which it is easy to derive the asset return correlations (26). Appendix 1 Integrating Yield Spread with Options Approach In this Appendix we show how the option pricing framework can be combined with the conventional yield spread approach to extract from yield spreads the cost of credit risk and recovery values. Consider Table 9.7, which lists U.S. Treasury and corporate bond yields. The above data can be transformed to derive the zero-coupon curves for the Treasury and the corporate bonds, as illustrated in Table 9.8. The zero-coupon table is used to derive the one-year forward rates N years forward (Table 9.9). This information is needed for CreditMetrics and other evaluation systems that are based on yield spreads. We can use the data for the zero-coupon curves to evaluate the implied parameter for a firm with a T- year bond outstanding. Suppose that company X has a two-year bond outstanding. This bond should offer a yield consistent with 6.25 percent per annum for a zero-coupon bond. In other words, by converting the two-

Page 390 TABLE 9.7 Prevailing Market Yields and Spreads Maturity (years) U.S. Treasury Par Yields Company X Par Yields Credit Spread 1 5.60% 5.85%.25 2 5.99% 6.34%.35 3 6.15% 6.60%.45 4 6.27% 6.87%.60 5 6.34% 7.04%.70 6 6.42% 7.22%.80 7 6.48% 7.38%.90 Note: Semiannual 30/360 yields. year corporate coupon bond into its two-year zero-coupon equivalent, its economic discrete time yield should be 6.25 percent. The risk-free yield on an equivalent two-year zero-coupon government bond is 5.91 percent. If we also assume that the bond has a face value of F = 100 at T = 2, its present value, B 0, should be B 0 = F/(1.0625) 2 = 88.58. TABLE 9.8 Zero-Coupon Curves Maturity (years) U.S. Treasury Par Yields Company X Par Yield 1 5.52% 5.76% 2 5.91% 6.25% 3 6.07% 6.51% 4 6.19% 6.80% 5 6.27% 7.18% 6 6.36% 7.37% 7 6.42% 7.54% Note: Continuously compounded 30/360 zero-coupon rates.

Page 391 TABLE 9.9 One-Year Forward Rates N Years Forward Maturity (years) U.S. Treasury Forwards Company X Forwards One-Year Forward Credit Spreads N Years Forward 1 5.52% 5.76%.24 2 6.30% 6.74%.44 3 6.40% 7.05%.65 4 6.56% 7.64% 1.08 5 6.56% 7.71% 1.15 6 6.81% 8.21% 1.40 7 6.81% 8.47% 1.65 Note: One-year continuously compounded 30/360. If the standard deviation of the rate of return on the firm's assets, σ, is equal to 20 percent, we can calculate the equity value S 0 for any given firm's value, V 0. The problem is to find V 0 and hence S 0 such that 88.58 + S 0 = V 0. This problem is equivalent to that of finding V 0 such that the put value is equal to: By trial and error, given the above assumptions, we find that by introducing V 0 = 144 into the Black- Scholes model, the derived equity value is S 0 = 55.44 such that 88.58 + 55.44 = 144.02 and also P 0 = 0.58. The cost of credit risk for the two-year corporate bond can be derived from the value of a put option on V 0 given F = 100, which is 0.58 for the above parameters. The present value of the recovery value of the loan is given by: By following the same procedure forσ=15 percent, we find that for V 0 = 124 the values of bond and equity are, respectively, 88.58 and 35.42. The cost of credit risk is P 0 = 0.57 and the present value of the recovery value is 81.5. Forσ=25 percent the yield

spread is consistent with a market value of V 0 = 170, which implies a leverage ratio, LR, of only 0.524 and an equity value of 81.42. Appendix 2 Risk-Neutral Valuation Using ''Risk-Neutral" EDFs Page 392 In Section 6 of this chapter we showed how to derive the "risk-neutral" EDFs from the actual EDFs. In this Appendix, we illustrate how to use these risk-neutral EDFs to value a bond or a loan that is subject to credit risk, and also how to value a credit derivative. (i) Case of a Single Cash Flow Example 1 Valuation of a zero-coupon bond with a promised payment in one year of $100, with a recovery of (1 LGD) if the issuer defaults. LGD is the "loss given default," which in this example is assumed to be 40 percent (Figure 9.A2.1). The default risk-free component, $100(1 LGD) is valued using the default risk-free discount curve, i.e.: where R denotes the one-year risk-free rate discretely compounded, assumed to be 10 percent, and PV is the abbreviation for present value. The risky cash flow is valued using the martingale approach, i.e.: where the expectation is calculated using the risk-neutral probability. The risk-neutral probability that the issuer might default one year from now is denoted as Q, and it is assumed to be 20 percent. Then:

Page 393 Figure 9.A2.1 Valuation of a Single Cash Flow Subject to Default Risk The present value of this zero-coupon bond subject to default risk is the sum of the default-free component and the risky component; i.e.: If the zero-coupon bond were default free, its present value would simply be its discounted value using the default-free interest rate; i.e.: We can then compute the implicit discount rate, Y, which accounts for default risk; i.e.: where CS denotes the credit spread. It is a solution of: Solving (5) for CS gives:

For this example, Y = 19.6 percent, so that the one-year credit spread for this issuer is 9.6 percent. (ii) Generalized Pricing Model for a Bond or a Loan Subject to Default Risk Page 394 The previous approach can be easily generalized to the valuation of a stream of cash flows [C 1,..., C i,... C n ]: or, in continuous time notation: where Q i denotes the cumulative "risk-neutral" EDF at the horizon t i and r i = ln(1 + R i ), is the continuously compounded discount rate for riskless cash flows. Example 2 What is the value of a 5-year bond with a face value of $100, which pays an annual coupon of 6.25 percent? The risk-free interest rate is 5 percent, the loss given default (LGD) is 50 percent, and the cumulative risk-neutral probabilities are given in the table below: Time Q i Discount Factor 1/(1 + R i ) ti Cash Flow PV 1 (Risk-Free Cash Flows) PV 2 (Risky Cash Flows) 1 1.89% 0.9512 6.25 2.97 2.92 2 4.32% 0.9048 6.25 2.83 2.71 3 6.96% 0.8607 6.25 2.69 2.50 4 9.69% 0.8187 6.25 2.56 2.31 5 12.47% 0.7788 106.25 41.37 36.21 Total 52.42 46.65 This methodology also applies to simple credit derivatives such as default puts, as we describe below. Example 3 What is the premium of a one-year default put that pays $1 if the underlying bond defaults?

Page 395 Assume a risk-neutral probability Q = 0.39 percent and an interest rate R = 5.8 percent; then: Appendix 3 Limitations of the Merton Model and Some Extensions The Merton (1974) model is appealing from an economic stand-point since it relates default to both the asset value process and the capital structure of the firm, together with the various debt covenants. In this framework default is never a surprise. So long as the asset value remains greater than the firm's liabilities, default should never occur in the immediate future (at least, with a probability materially different from zero). 27 The limitations of the Merton model are related to the practical difficulties in implementing it: The asset value and its volatility are not directly observable and are hard to estimate. KMV, however, proposes an elegant methodology with which to derive the implied asset values and asset volatility from equity prices (see Section 5). The risk-free interest rates are constant and therefore the relationship between interest rate risk and equity risk cannot be modeled. But this assumption has been relaxed by several authors. (See, e.g., Kim, Ramaswamy, and Sundaresan 1993; Shimko, Tejima, and van Deventer 1993; and Anderson and Sundaresan 1996.) Reliance on the absolute priority rule to describe the payoff to debt holders in default makes the valuation of coupon debt difficult, as a failure to pay a coupon may also trigger default. Thus, defaultable coupon debt can be

viewed as a compound option. The value of the debt depends on whether the previous coupon payment passed without incident (see Geske 1977). Page 396 For complex capital structures, the valuation of a risky bond depends on the value of the senior debt. Junior debt is a contingent claim on senior debt that substantially complicates valuation. In practice seniority rules are not fully enforced and in many bankruptcy cases we observe a transfer of value from senior claim holders to junior claim holders and equity holders. 28 Several recent contributions based on Merton's framework abandon this concept of interdependence in default in favor of simultaneous default or "cross-default" (see Section A3.2). A3.1 Empirical Evidence Jones, Mason, and Rosenfeld (1984) adopted a simple capital structure and compared Merton's model to the "na ve" method of discounting cash flows at the risk-free rate. They found that the model outperforms the na ve approach for non-investment-grade bonds, but not for investment-grade securities. According to this study, Merton's model consistently prices bonds higher than the traded price; i.e., implied credit spreads from Merton's model are too low. These somewhat negative results may be due to several factors: The most serious is the violation of the seniority rules (see Footnote 27). The choice of a capital structure that is too simplistic, as a proxy for the actual capital structure. Deterministic interest rates. Kim, Ramaswamy and Sundaresan (1993) show that the introduction of stochastic interest rates in Merton's framework significantly improves the performance of the model. The poor quality of bond data and calculated yield, which do not correctly adjust for call features. Sarig and Warga (1989), using a new data set provided by Lehman Brothers, found that for corporate zero-coupon bonds, spreads seem to be qualitatively close to the Merton model predictions.

Page 397 Delianedis and Geske (1998) use the option pricing model of Merton (1974) and Geske (1977) to estimate risk-neutral probabilities of default during the period 1987 to 1996. They show that these probabilities contain forward-looking information about rating migration and default. In addition, the short-term default probability from Geske's model is able to predict imminent cash flow problems. A3.2 Extensions of Merton's Model Geske (1977) extended Merton's model by showing that multiple default options for coupons, sinking funds, junior debt, safety covenants, or other payment obligations could be treated as compound options. In Geske's model default may occur before maturity, at a coupon payment date, even though the asset value is greater than the coupon payment that is due to be made. This happens when the value of equity is close to zero after the coupon payment is made and the shareholders are therefore better off not paying the coupon. Black and Cox (1976) reinterpreted default as the event that occurs when the asset value V hits some boundary K at any time, and not just at maturity or at a coupon payment date. This assumption is consistent with a bond covenant that specifies some net-worth constraint. For example, if the value of the firm were to drop below a prespecified level K, then the bondholders can call the firm into paying the debt immediately, or alternatively force the firm to cross-default on all its debt at once if it is insolvent. 29 But, in this framework, financial distress is not explicitly modeled in the pricing of corporate debt. 30 Kim, Ramaswamy, and Sundaresan (1993) propose a Merton-type model with stochastic interest rates where financial distress arises out of cash flow constraints, and bankruptcy is costly. Anderson and Sundaresan (1996) extend the previous model by allowing the firm to make a "take-it-or-leave-it" offer to the debtholders. The sharing of values by the debt holders and shareholders is obtained endogenously. The creditors have the choice of accepting or rejecting the debt service (which is bounded by the cash flows generated by the firm). If the debt service is rejected, the firm is liquidated and the stakeholders incur liquidation costs. This model

Page 398 generates spreads that are consistent with the observed spreads in the market, even at very low levels of debt and liquidation costs. Longstaff and Schwartz (1995) allow deviations from the absolute priority rule, but the allocation of the value of the bankrupt firm amongst the various stakeholders is specified exogenously rather than being the outcome of an endogenous process of negotiation as in Anderson and Sundaresan (1993). Longstaff and Schwartz introduce three key assumptions: Default occurs when the asset value falls below a constant threshold K as in Black and Cox (1976). Default occurs simultaneously for all securities. Therefore, there is no need in their framework to model the bankruptcy process. The recovery rate is constant for each security and can be interpreted as the outcome of some complex post-bankruptcy bargaining process (exogenous to the model) among the various claimants. The risk-free interest rate is stochastic and follows a simple one-factor Vasicek process with mean reversion, which is correlated with the asset value process: whereβand m denote the mean reversion parameters of the interest rate process,ρis the correlation between the processes for the default-free interest rate and asset value, and Z r and Z V are standard Brownian motions. Longstaff and Schwartz make use of numerical approximation to compute the probability of default happening before the maturity of the debt. The Vasicek process for the default-free interest rate allows closed-form solutions for risky zero-coupon bonds. Coupon bonds can be valued as a portfolio of zero-coupon bonds because the event of default is not specific to a particular bond. They find that the further away the asset value is from the default boundary, the lower is the credit spread. As time increases,

Page 399 the probability of default increases. Credit spreads can be monotone increasing, or hump-shaped, which is consistent with empirical evidence. Credit spreads increase with any increase in correlation between assets and interest rates. 31 The model produces a negative correlation between the level of Treasury yields and credit spreads, which is consistent with Merton's model. While some aspects of the model are oversimplified, the model can be implemented in practice. Parameters can be estimated from term structure data and from historical firm data. Longstaff and Schwartz's model has been extended by Saa-Requejo and Santa-Clara (1997). They suggest that the boundary value K should represent the present value of the liabilities rather than being a fixed number. 32 Leland (1994) and Leland and Toft (1996) introduce costly bankruptcy and tax deductibility of interest rate charges. Shareholders then choose leverage to maximize the value of equity by selecting the best trade-off between tax benefits and bankruptcy costs. Bankruptcy is thus endogenized. It occurs when it ceases to be in the interest of shareholders to give up $1 to pay the next $1 of interest rate payment. This model provides useful insight but in its current form it is too restrictive to be of practical use. Notes 1. KMV is a trademark of KMV Corporation. The initials KMV stand for the first letter of the last names of Stephen Kealhofer, John McQuown, and Oldrich Vasicek, who founded KMV Corporation in 1989. S. Kealhofer and O. Vasicek are two former academics from U.C. Berkeley. 2. See Moody's (1995). See also Chapter 7 for a description of the Moody's data. 3. This can lead to adverse selection of corporate customers in banks. Indeed, if the pricing of loans is based on this average historical default rate, then a typical customer will be overcharged and may have an incentive to leave, while the worst obligors in the class will benefit from an advantageous pricing with regard to their actual credit risk. 4. See, for example, Duffie (1992). 5. This contribution has been followed by Galai and Masulis (1976); Black and Cox (1976); Merton (1977); Lee (1981); Ho and Singer

Page 400 (1982); Pitts and Selby (1983); Johnson and Stulz (1987); Chance (1990); Cooper and Mello (1991); Shimko, Tejima, and van Deventer (1993); Leland (1994); and Kim, Ramaswamy, and Sundaresan (1996). 6. In the following "exogenous" refers to the assumptions specified outside the model and that constitute a given in the model derivation, while "endogenous" characterizes results derived from the model. 7. See also Litterman and Iben (1991) and Madan and Unal (1996). This approach is consistent with the CreditMetrics methodology suggested by JP Morgan (1997) to assess credit risk exposure for the banks' portfolio of fixed-income instruments. 8. This approach is difficult to implement in practice, and in some instances the model calibration yields negative default probabilities. This inconsistent result comes from the fact that liquidity risk is ignored (at least in early versions of this approach) and the recovery factors not only vary over time, but also should be endogenously determined in the model since the loss incurred by the debt holders should depend explicitly on the value of the firm's assets. 9. Sections 2 to 4 are drawn from our paper "Credit Risk Revisited," which appeared in Risk, March 1998. 10. The model builds on Black and Cox's (1976) extension of Merton's (1974) model. 11. A leverage factor equal to 0.7 is obtained by a face value F = 77. 12. The computed cost of default is slightly different from the put value due to rounding errors. 13. See Vasicek (1997) and Kealhofer (1995,1998). See also Appendix 1 of Chapter 8. 14. That is, the composition of its liabilities: equity, short-term and long-term debt, convertible bonds, and so on. 15. Financial models consider essentially market values of assets, and not accounting values, or book values, which only represent the historical cost of the physical assets, net of their depreciation. Only the market value is a good measure of the value of the firm's ongoing business, and it changes as market participants revise the firm's future prospects. KMV models the market value of assets. In fact, there might be huge differences between both the market and the book values of total assets. For example, as of February 1998 KMV has estimated the market value of Microsoft assets as $US228.6 billion versus $US16.8 billion for their book value, while for Trump Hotel and Casino the book value, which amounts to $US2.5 billion, is higher than the market value of $US1.8 billion.

16. The exception is when the firm's portfolio of businesses has changed substantially through mergers and acquisitions, or restructuring. 17. See also Crouhy and Galai (1994); Bensoussan, Crouhy, and Galai (1994,1995); and Vasicek (1997) for the valuation of equity for more complex capital structures which, for example, include equity warrants and convertible bonds. Page 401 18. In the general case, the resolution of this model may require the implementation of complex numerical techniques, with no analytical solution, due to the complexity of the boundary conditions attached to the various liabilities. See, for example, Vasicek (1997). 19. It can be shown thatσ S =η S,A σ whereη S,A denotes the elasticity of equity to asset value; i.e.,η S,A = (V/S) ( S/ V) (see Bensoussan, Crouhy, and Galai 1994). In the simple Merton framework, where the firm is financed only by equity and zero-coupon debt, equity is a call option on the assets of the firm with a striking price of the face value of the debt, and a maturity of the redemption date of the bond. Then, the partial derivative S/ V is simply the delta of the call with respect to the underlying asset of the firm. 20. See the Appendix for Chapter 8: the distance to default can be easily derived from expressions (1) and (4). 21. See, for example, Jarrow and Turnbull (1997b), Chapters 5 and 6. 22. An empirical issue is whether the corporate spread should be calculated over the Treasury curve, or instead over the LIBOR (swap) curve. It seems that the best fits are obtained when spreads are calculated over LIBOR. 23. See, for example, Elton and Gruber (1995, Chapter 8). While a multifactor model can be implemented directly, the model gains very convenient mathematical properties if the factors are uncorrelated, i.e., orthogonal. There are simple techniques to convert any set of correlated factors into a set of orthogonal factors. In that case we would have cov(i 1, I 2 ) = 0. 24. For a review of multifactor models see Elton and Gruber (1995) and Rudd and Clasing (1988). The most widely used technique in portfolio management is the single-index model, or market model, which assumes that the comovement between stock returns is due to a single common index, the market portfolio. The return generating process for this model is described by r k =α k +β k r M +ε k, where r M denotes the rate of return on the market portfolio. This model can then be extended to capture industry effects beyond the general market effects. Rosenberg (1974) has developed a model for predicting extra market covariance which relates not only industry

Page 402 factors, but also company-specific descriptors such as market variability, and which captures the risk of the firm as perceived by the market, earnings variability, index of low valuation and lack of success, immaturity and smallness, growth orientation, and financial risk (see Rudd and Clasing 1988). Finally, a number of multifactor models have been proposed which relate security returns to macroeconomic variables. Excess returns are explained by the unexpected changes, or innovations, in variables such as inflation, economic growth as measured by unexpected change in industrial production, business cycles as proxied by the corporate spread over Treasuries, longterm interest rates, short-term interest rates, and currency fluctuations (see Chen, Roll, and Ross 1986 and Berry, Burmeister, and McElroy 1988). 25. Compustat is a database of financial and economic information about firms. 26. SIC denotes the Standard Industrial Classification, which is a U.S.-based business classification system. 27. This is in contrast with the reduced-form approach presented in the next chapter, where default is modeled as an exogenous Poissontype process independent of the capital structure of the firm and its asset value. In this framework default is always a ''surprise" and does not relate to the financial health of the firm. 28. Frank and Torous (1989) found that substantial deviations from strict priority rules are common in the United States (Chapter 11). Essentially, all the observed deviations were in favor of equity holders and against the existing claims of debt holders. They estimate the impact of these deviations on the yield spread predicted by Merton's model as up to 120 to 150 bp for a 15-year bond. See also Kim, Ramaswamy, and Sundaresan (1993) and Anderson and Sundaresan (1996). 29. If we assume that shareholders do not have to put additional money into a firm to allow it to continue operating, then it is not in their interest to shut down the firm when the asset value falls below the threshold K. Indeed, when the assets value falls below K before debt maturity, there is still a positive probability that the firm will recover before the debt matures. Therefore, the value of equity (debt) is worth more (less) if the firm is not shut down. 30. See also Chance (1990). 31. In practice the correlation ρ is negative. 32. See also Hull and White (1995).